More than "could be suggested".
From "Of the Finding of the Ring" in the Prologue to the Second Edition of LOTR:
Gandalf, however, disbelieved Bilbo’s first story, as soon as he heard it, and he continued to be very curious about the ring. Eventually he got the true tale out of Bilbo after much questioning, which for a while strained their friendship; but the wizard seemed to think the truth important. Though he did not say so to Bilbo, he also thought it important, and disturbing, to find that the good hobbit had not told the truth from the first: quite contrary to his habit. The idea of a ‘present’ was not mere hobbitlike invention, all the same. It was suggested to Bilbo, as he confessed, by Gollum’s talk that he overheard; for Gollum did, in fact, call the ring his ‘birthday present’, many times. That also Gandalf thought strange and suspicious; but he did not discover the truth in this point for many more years, as will be seen in this book
And from "The Shadow of the Past":
‘Let me see – it was in the year that the White Council drove the dark power from Mirkwood, just before the Battle of Five Armies, that Bilbo found his ring. A shadow fell on my heart then, though I did not know yet what I feared. I wondered often how Gollum came by a Great Ring, as plainly it was – that at least was clear from the first. Then I heard Bilbo’s strange story of how he had “won” it, and I could not believe it. When I at last got the truth out of him, I saw at once that he had been trying to put his claim to the ring beyond doubt. Much like Gollum with his “birthday present”. The lies were too much alike for my comfort. Clearly the ring had an unwholesome power that set to work on its keeper at once.