The former Raith Rovers and Dundee frontman moved to Firhill last summer but has failed to net in 17 outings for the Jags, the vast majority of which have come off the bench.
Baird was used mostly as Kris Doolan's understudy during the first half of the season but has now dropped even further down Thistle's pecking order following the arrival of on-loan Sheffield United striker Lyle Taylor earlier this month.
The 28-year-old is now set to try his luck in the Scottish Championship after former St Mirren midfielder Ricky Gillies - who now represents the player on behalf of the Pro-Legal Group - confirmed he had parted company with Partick.
He told Press Association Sport: "John has agreed a compromise agreement with Thistle this morning. They have paid up his contract and allowed him to move on.
"He hasn't got anything else lined up just yet but there is interest from a few Championship clubs."