The Scotland captain is expected to make the switch to North London in the summer - if they avoid the drop
Out of contract Liverpool star Andy Robertson "would go to Celtic" if an offer came in with a summer switch to Tottenham in jeopardy.
The Scotland captain has announced he will bring the curtain down on a glittering nine years with the Reds where he has lifted every major honour possible - including two Premier League titles.
Reports suggest that the 32-year-old has "verbally agreed" to join Tottenham - who showed interest in the left-back in January - provided they avoid relegation from the Premier League.
After losing Roberto De Zerbi's first game in charge, Spurs are currently two points from safety with just six games left to save their season.
Should the deal fall apart, Mulgrew - who was international teammates with Robertson - reckons that the defender would jump at the chance to join the Hoops, if an offer came in.
He told talkSPORT: “I think he would go to Celtic. I don’t want to speak for him, but I’d imagine being a Celtic fan is a great opportunity for him at 32.
“He’s still got a lot to give, but I don’t think can Celtic afford to put a budget in two left backs of Kieran Tierney and Andy Robertson.
“I don’t imagine Celtic will want to pay the two left backs the biggest wages. I think if Celtic are going to put wages like that towards a player, it’ll be a forward area.”
Newcastle United legend Alan Shearer - who was relegated while in charge of the Magpies in 2009 - admits he has changed his view on Spurs' fate following last weekend's loss to The Black Cats.
The Match of the Day pundit told Betfair: "I have changed my opinion on Spurs being relegated, because I look at the other clubs in and around them and how they are.
"I look at West Ham, who are fighting and scrapping. I look at Leeds, who got a great result and are fighting and scrapping. I look at Forest, who got a point and who are fighting and scrapping.
"I don't see that with Tottenham. We were looking for the manager bounce in the first game and clearly that didn't happen. It looks as if Romero's going to be out for the rest of the season, which is a blow.
"As we sit here now, Tottenham are the team that are going down because I don't see enough fight. I don't see enough attitude, I don't see enough desire to try and keep them in the Premier League, whereas the teams in and around them are showing exactly that."
Sports finance expert Professor Rob Wilson has warned Tottenham of the financial impact facing the club should they drop to the second tier of English football with the club paying out a £2.63m gross per week wage bill - the seventh highest in the Premier League.
Wilson told The i Paper: “Some other clubs could even have 90 per cent relegation clauses or agreements for players to move.
“But because Spurs have been relatively stable in the Premier League, they don’t have the clause they would need in order to properly survive.
“It’s nowhere near enough. You need a minimum 75 per cent relegation clause in order to balance your books when you go down.”