The ex-politician was jailed for three years in 2011 after he was found guilty of lying under oath during his successful defamation action against the now-defunct News of the World newspaper.
Mr Sheridan, 50, has always maintained his innocence and today lodged an application to the Scottish Criminal Cases Review Commission (SCCRC) based in Glasgow.
He said his lawyer Gordon Dangerfield had been working for a year-and-a-half on a dossier which he said "we are very, very confident is going to lead eventually to the quashing of my criminal conviction".
Speaking outside the SCCRC office, Mr Sheridan said: "We are confident it is going to be the start of the journey to clear my name and also give some succour and support to those who have stuck by me over many years."
Mr Dangerfield said: "It is true, as Tommy says, that we believe we have uncovered a very widespread, very extensive criminal conspiracy against Tommy Sheridan and against the course of justice in his civil trial, his criminal trial and in fact ongoing to this day.
"I think people will be shocked when the full story comes out about how high this conspiracy goes and the people who have been involved in it, including people who held and in some cases still hold very high offices."
In a 2004 article, the News of the World reported Mr Sheridan had visited a swingers club and cheated on his wife Gail. He was an MSP representing the Scottish Socialist Party at the time.
He was awarded £200,000 in damages after winning his defamation case against the tabloid but ended up on trial for perjury and was convicted at the end of 2010.
Sheridan - who was freed from prison after serving just over a year of his three-year sentence - will urge the SCCRC to look at his case and consider referring it back to the High Court for an appeal.
Senior judges refused him leave to appeal against the conviction in 2011. His legal team wanted to argue that he had been denied a fair trial because of the amount of publicity generated before it got under way.
In cases where there has already been an unsuccessful appeal or leave was previously refused, the only route back to appeal judges is via the SCCRC.
The commission will first have to decide whether to accept the application. If it does, it then has to determine whether there might have been a miscarriage of justice, a process which usually takes a few months.
If the commission decides there might have been such a miscarriage, it would then refer it back to the High Court, and the case proceeds like a regular appeal.
Mr Dangerfield, of Archer Coyle solicitors, said the SCCRC has thousands of documents to consider and he would not expect a decision for a number of months.
The application focuses on two main areas, he said.
The first is the alleged "wrongful judicial exclusion of evidence" which the lawyer suggests prevented the "conspiracy" coming to light.
Mr Dangerfield said: "The other is that there has now been a huge amount of further information uncovered which gives us new evidence grounds of appeal that there was this extensive conspiracy to pervert the course of justice in both of Tommy's trials."