No.1:  “At that moment there was a knock on the door, and Sam came in. He ran to Frodo and took his left hand, awkwardly and shyly. He stroked it gently and then he blushed and turned hastily away.

`Hullo, Sam!' said Frodo.

`It's warm!' said Sam. `Meaning your hand, Mr. Frodo. It has felt so cold through the long nights. But glory and trumpets!' he cried, turning round again with shining eyes and dancing on the floor. 'It's fine to see you up and yourself again, sir! Gandalf asked me to come and see if you were ready to come down, and I thought he was joking.' ” 

--------------------from The Fellowship of the Ring 

(This is the point where my suspicion first arises. Here Sam acts, as Tolkien has described, more like a lad (or lass) in love than a servant and journey companion.)

No.2: “Gollum disappeared. He was away some time, and Frodo after a few mouthfuls of lembas settled deep into the brown fern and went to sleep. Sam looked at him. The early daylight was only just creeping down into the shadows under the trees, but he saw his master's face very clearly, and his hands, too, lying at rest on the ground beside him. He was reminded suddenly of Frodo as he had lain, asleep in the house of Elrond, after his deadly wound. Then as he had kept watch Sam had noticed that at times a light seemed to be shining faintly within; but now the light was even clearer and stronger. Frodo's face was peaceful, the marks of fear and care had left it; but it looked old, old and beautiful, as if the chiselling of the shaping years was now revealed in many fine lines that had before been hidden, though the identity of the face was not changed. Not that Sam Gamgee put it that way to himself. He shook his head, as if finding words useless, and murmured: `I love him. He's like that, and sometimes it shines through, somehow. But I love him, whether or no.'”
--------------------from The Two Towers

(The second point is not as shocking as the first one. A man who speaks "I love him" may not necesarrily be  out of romantic affection. Admiration can be the motive, too. But still it’s a bit unusual. ) 

No.3: “He was naked, lying as if in a swoon on a heap of filthy rags: his arm was flung up, shielding his head, and across his side there ran an ugly whip-weal. `Frodo! Mr. Frodo, my dear!' cried Sam, tears almost blinding him. `It's Sam, I've come!' He half lifted his master and hugged him to his breast. Frodo opened his eyes.

`Am I still dreaming?' he muttered. `But the other dreams were horrible.'

`You're not dreaming at all, Master,' said Sam. `It's real. It's me. I've come.'

`I can hardly believe it,' said Frodo, clutching him. `There was an orc with a whip, and then it turns into Sam! Then I wasn't dreaming after all when I heard that singing down below, and I tried to answer? Was it you?'

'It was indeed, Mr. Frodo. I'd given up hope, almost. I couldn't find you. '

'Well, you have now, Sam, dear Sam, ' said Frodo, and he lay back in Sam's gentle arms, closing his eyes, like a child at rest when night-fears are driven away by some loved voice or hand. Sam felt that he could sit like that in endless happiness; but it was not allowed. It was not enough for him to find his master, he had still to try and save him. He kissed Frodo's forehead. `Come! Wake up Mr. Frodo!' he said, trying to sound as cheerful as he had when he drew back the curtains at Bag End on a summer's morning.
------------------ from The Return of the King. 

(Well, this part contains the most concentrated effusion of the strong attachment between Sam and Frodo. "My dear" is usually used by people to call someone younger than he/she. But Frodo is actually older than Sam (Frodo is in his fifties, and Sam is practically a lad). "Endless happiness" almost exclusively belongs to lovers.)  

Given the fact that the Lord of the Rings devotes only a paltry amount of description to love and sex (The exalted matrimony between Aragorn and Arwen, which involves the sacrifice of immortality, barely gets any page in the book), the words and sentences used to depict Sam and Frodo seem disproportionately sumptuous. But still their relationship remains dubious yet uncertain. The man who has the biggest authority to answer the question has passed away more than two decades ago, so it is impossible to get any definitive revelation on this matter, as J.K. Rowling would do about Dumbledore. Whether yes or no, I would love the Lord of the Rings as always. I cannot forget the moment when I was reading the conclusion of the whole epic tale (starting from the Hobbit): 

“But to Sam the evening deepened to darkness as he stood at the Haven; and as he looked at the grey sea he saw only a shadow on the waters that was soon lost in the West. There still he stood far into the night, hearing only the sigh and murmur of the waves on the shores of Middle-earth, and the sound of them sank deep into his heart. Beside him stood Merry and Pippin, and they were silent.

At last the three companions turned away, and never again looking back they rode slowly homewards; and they spoke no word to one another until they came back to the Shire. but each had great comfort in his friends on the long grey road.

At last they rode over the downs and took the East Road, and then Merry and Pippin rode on to Buckland; and already they were singing again as they went. But Sam turned to Bywater,and so came back up the Hill, as day was ending once more.And he went on, and there was yellow light, and fire within; and the evening meal was ready, and he was expected. And Rose drew him in, and set him in his chair, and put little Elanor upon his lap.
He drew a deep breath. 'Well, I'm back,' he said.”

And I heave a sigh. The short sentences, punctuated by a string of ands, convey some unspeakably complex feelings about triumph and relief and loss and sadness.