A major shift towards same-sex marriage is under way within the Liberal Party, with marriage equality advocates saying 11 Liberal MPs have privately switched to supporting a change since January.

Liberal MPs could discuss whether they should be granted a free vote on the issue as early as Tuesday, with Liberal Democrat senator David Leyonhjelm due to begin the parliamentary debate on his "freedom to marry" bill on Thursday.

So far, only a small group of Liberal MPs, including frontbenchers Malcolm Turnbull, Simon Birmingham and Kelly O'Dwyer, have come out publicly in favour of same-sex marriage.

But marriage equality advocates, who have been meeting with MPs over recent weeks, said there is now a significant change under way inside the party.