The left-back is out of contract in the summer and is unlikely to sign a new deal to extend his stay at Firhill.
After playing a key role in the Jags' First Division title success last term, Taylor-Sinclair has impressed in the opening weeks of the Premiership campaign, with champions Celtic credited with an interest in the 22-year-old.
Thistle would not be due compensation if he leaves on a Bosman, and boss Archibald says the Jags are fighting a losing battle to retain the former Montrose youngster.
He said: "He has kicked on again this year. He came back really sharp in pre-season and he has improved a lot of aspects of his game.
"He is a good height and he has filled out. When he came initially from Montrose he was a wee, scrawny lad, but you can see now he is an athlete and very quick. The modern-day full-back is more or less a winger, trying to get them to defend is not easy!
"He is out of contract at the end of the season and I don't think we will keep him. We offered him a new deal in the summer.
"Aaron is a good lad, he has been brilliant here. But I think he wants to go and test himself at a higher level and a bigger club and nobody will stand in his way.
"I don't think we are (due compensation). He is an April birthday, he is 23 in April. If he had been born a couple of months later we would have been due something so it is a sore one. We have had three good years out of Aaron, he has progressed and done well for the club."
Archibald's side head to Pittodrie on Monday night aiming to upset in-form Aberdeen and record their fourth top-flight success of an encouraging campaign so far.
The Dons clash is the latest Jags fixture to be rescheduled for live television coverage, but Archibald admits the timing is far from ideal for players or supporters as the Firhill side get set for a trip to the North East.
He said: "I said last week there is certainly a market for it on different nights of the week. We have had great success with Fridays, but I think they should regionalise it where they can. It would be much better. I am a big fan of Friday nights.
"But it would be a lot easier, if I was a fan, going to the games if there wasn't as far to travel. It is the same for the Aberdeen fans.
"It is hard to complain about it because that is what we won promotion for. We are on TV and it is great.
"The new fixtures came out and we haven't been changed, which is great. We are back in normal days for training and it is a normal week.
"Players like that, they like to be in a habit. We are delighted to be on TV but we are glad to be back playing Saturday football for the next couple of months."
Aberdeen goalkeeper Jamie Langfield claims the buzz of anticipation flowing round Pittodrie is as strong as he has known.
The former Dundee stopper is now in his eighth year with the Dons, but says he has not felt the kind of optimism generated by new boss Derek McInnes' efforts since Jimmy Calderwood led them to a glamour clash with German giants Bayern Munich back in 2008.
Langfield said: "I've played here when things were really good under Jimmy Calderwood and I've been here during the bad times as well. But I would say the way things are feeling right now, it's up there with the best I've known."