Archibald was on Swindon Town's managerial shortlist last season and has been linked with other vacancies.
Osman says the loss of Archibald - who guided Thistle to a top six finish last season - would be a hard act to follow.
"Since I signed, he's only got better, every season. It's no surprise teams are interested in him," Osman said.
"We don't want him to go anywhere. He's probably one of the best managers I have played under.
The boys love him. We just pray he stays here for another season."
Osman's Partick Thistle team-mate Adam Barton echoes his captain's sentiments.
"I would be personally disappointed because what he has done is really good," the midfielder said.
"As a footballer, you like managers who really stick by their players. You come across so many managers who chop and change things and I have been through that many managers that many times.
"New managers come in and they don't even want to see you play, they just want to bring their own players in and chop and change. To see him go would not be a good thing."
Partick Thistle are on the verge of losing promising centre-back Liam Lindsay, who has travelled to Barnsley for a medical following the clubs agreeing a fee for the player.
Subject to the completion of the medical, Lindsay will sign a three-year deal at Oakwell.

Deborah Steel, who was 37 and ran the Royal Standard in Ely, was last seen in the early hours of 28 December. Her body has not been found.
No further action will be taken against a 50-year-old and a 70-year-old, both from Ely, Cambridgeshire Police said.
A 72-year-old man from Ely has been re-bailed until 17 February.
Ms Steel's disappearance was recently reclassified from a long-term missing person inquiry to a murder inquiry by officers.

The former Hearts captain recently left Leeds, having moved to Elland Road last summer despite a trial at Rangers.
"I'm excited to come here from Leeds and on the first day it felt like it was the right decision," Zaliukas told the club's website.
"It's a brilliant stadium and it's going to be fun to play here."
Zaliukas played more than 200 times for Hearts over seven years, but made just 15 appearances for Leeds last term.
His first competitive match for Rangers could be against Hibs in the Challenge Cup on 5 August.
The Ibrox outfit's first Scottish Championship fixture of the campaign is against Hearts at Ibrox on 10 August.
"I think the Championship is going to be a very interesting league because Hearts and Hibs are in here, and even Falkirk and Queen of the South are going to be tough teams," the 30-year-old added.
"I think our squad could compete straight away in the Premiership, everyone is experienced and top players. I think we are going to have a very good squad."
Rangers have also added former St Mirren defender Darren McGregor to their ranks this summer.
Partick Thistle captain Abdul Osman says the players are praying manager Alan Archibald stays at the club for another season.
[NEXT_CONCEPT]
Two of the three men arrested by detectives investigating the disappearance of a Cambridgeshire pub landlady in 1997 have been released.
[NEXT_CONCEPT]
Lithuanian defender Marius Zaliukas has signed a two-year contract with Rangers, after impressing during a trial with the Ibrox club.