The Swiss shot-stopper is a free agent after leaving Tannadice at the end of his contract and has been a transfer target for Hoops manager Ange Postecoglou.
The Hoops require extra competition for ex-Manchester City and Tottenham No1 Joe Hart after frozen out Vasilis Barkas moved to Eredivisie side FC Utrecht on a season-long loan.
Scott Bain is established as the main back-up option but Postecoglou is now expected to add Siegrist to his squad for next season.
The 30-year-old, who kept 13 clean sheets last term, spent four years with the Arabs and a series of impressive performances between the sticks have caught the eye of the Parkhead boss.
His consistent form has alerted a host of clubs in England, with Championship sides Preston North End and Ipswich Town also chasing his signature.
According to the Scottish Sun, a transfer to Celtic is close to completion with Siegrist set to become the club’s newest addition.
Siegrist began his youth career with FC Basel in his homeland before joining Aston Villa as a teenager in 2009.
He failed to make his first-team breakthrough at the Midlands club and a move to Vaduz followed before arriving in Scotland in June 2018.
The Glasgow giants have held positive talks with his representatives and with domestic and European commitments next season, Postecoglou will be delighted to have three experienced keepers to call upon.
Youngster Conor Hazard remains on loan at HJK Helsinki until January 2023, which leaves reserve keeper Tobi Oluwayemi as the only other option.
However, the imminent arrival of Siegrist could allow the 19-year-old England youth international to head out on loan.