Welsh has been a standout for Jackie McNamara's side since making a temporary switch along the M8 earlier this year, an arrangement that was extended to allow him to see out the remainder of the Irn-Bru First Division campaign.
The Jags rounded off their season with a 2-2 draw at Hamilton Accies on Saturday, with Welsh netting his second goal in four games to clinch a point.
The 22-year-old is out of contract at Easter Road this summer and available on a Bosman transfer.
Thistle are still awaiting an answer from midfielder Paul Cairney over whether he will sign on the dotted line for another season.
But McNamara is keen to land Welsh either way as the midfielder ponders his options.
He said: "I hope that is not his last game for us. We have made him an offer and hopefully he accepts it. I don't know about his situation at Hibs, but we certainly want to keep him because he has been excellent for us.
"He showed a lot of desire to go and get his goal on Saturday. He is only 22, but he is a real asset for us."
Welsh is not the only player on McNamara's radar as he attempts to bolster his squad for a fresh challenge at the First Division title in his second season in the dugout.
The Jags boss is keen to add to his playing pool sooner rather than later and SportTimes understands Annan Athletic youngster Aaron Muirhead is one of the names on his shopping list.
The 21-year-old can play in both defence and attack and has proved his worth for Harry Cairney's side in the Third Division this term.
With finance at Firhill tight, McNamara knows he must act quickly to tie up his intended targets and is eager to keep up the pace after agreeing new deals with captain Alan Archibald, defensive duo Aaron Sinclair and Conrad Balatoni and strike star Mark McGuigan in recent weeks.
He said: "We are working on a few things at the moment and we are trying to do things early.
"Funds are not great so we have to make sure we get in there early and try and bring players to the club."
There have already been exits from Firhill, too, with striker Tommy Stewart told that his contract will not be renewed after a disappointing second season with the Jags.
Youngsters Gavin Griffin and Kieran Burns have also been told they are free to leave.
McNamara said: "It has been a frustrating year for Tommy. He has ability but sometimes it just doesn't work out.
"Hopefully he can kick start his career because he has the potential to be very good."