The case of the dog Laika demonstrates that the Soviet Space Program could and did send living things on missions that were known beforehand to be fatal due to political pressure, which is likely one source of this rumor.
With the success of Sputnik I, Krushchev wanted to keep the momentum and public interest going by sending a living animal into orbit by a notable anniversary of the Bolshevik cause, leaving them with mere months to prepare. The Soviet scientists knew that there was no way that they could launch and safely retrieve any object from orbit at the time, so Laika was always going to die, they just didn't know how. As it turned out, the satellite they prepared for Laika was not insulated enough, and she most likely died due to extreme heat, very soon after launch.
The Soviet government did lie about her death, stating she survived for longer than she did and that she died for different reasons (the original plan was to euthanize her after a few days with poisoned food). Since her body was never recovered, the actual cause of death can't be definitively stated, but based on the data they received from the satellite, it was likely due to heat.
Upon learning that there was no survival scenario planned for Laika, world sentiment towards the Sputnik II mission became hostile, and sparked a discussion about the ethics of animal testing. Although within the USSR there was very little publicly written against Sputnik II, again likely due to censorship and political pressure, years later, after the fall of the Soviet Union, one of the scientists stated:
"Work with animals is a source of suffering to all of us. We treat them like babies who cannot speak. The more time passes, the more I'm sorry about it. We shouldn't have done it ... We did not learn enough from this mission to justify the death of the dog."
So we know there was political pressure that forced the scientists to construct a mission they otherwise probably would not have done in that amount of time in that way. And we know that the USSR experienced negative press about the event afterwards. I find it extremely unlikely, given the experience of Laika and Sputnik II, that any Soviet premier would force the space program to make similar shortcuts for a manned mission, and the scientists would be very conscious of the world attention towards their mission. That doesn't rule out accidents, of course, but it seems extremely unlikely that such an event could have been buried for years and not come to light after the end of the USSR.
Just wanted to point out this: the fact that Gagarin is in orbit was broadcasted to the world even before he returned.
The call about a rocket launch from USSR reached the White House at 1:35 am. The Soviet news agency, TASS, announced the news to the world half an hour later. By 5 am, the news of Gagarin's safe return were broadcast.
At the time of the TASS announcement, it wasn't clear if Gagarin would return safely or not.
I can't say how it started but the rumor was apparently already well-known before Gagarin's flight since it was mentioned in this news article announcing Gagarin's success and included mention of Russian denials.
EDIT: So you made me curious and some googling turned up this article from Robert Heinlein (see page 60, starting at the bottom of the first column) about an encounter on a trip to the Soviet Union in 1960 as a possible source for the theory. He claims he was told by Russian cadets about a manned rocket launch that happened:
Shortly one of the cadets asked us what we thought of their new manned rocket? We answered that we had had no news lately — what was it and when did it happen? He told us, with the other cadets listeningand agreeing, that the rocket had gone up that very day, and at that very moment a Russian astronaut was in orbit around the earth — and what did we think of that...
He continues:
This is the rocket the Soviets tried to recover and later admitted that they had had some trouble with the retrojets; they had fired while the rocket was in the wrong attitude. So WHAT is the answer? Did that rocket contain only a dummy, as the pravda now claims? Or is there a dead Russian revolving in space? — an Orwellian "unperson," once it was realized he could not be recovered.