The versatile Bhoys star has his eyes on a new contract despite limited game time this season.
A combination of injuries and increased competition for places, such as Kieran Tierney's emergence at left-back, have limited the 30-year-old to just 20 appearances in all competitions this season at club level.
That follows a disappointing campaign last term in which Mulgrew played just eight times in the Scottish Premiership, and with his contract expiring at the end of June it had seemed that the defender was set to leave to Parkhead.
However, the Scottish international, who has played over 200 times for the club since moving from Aberdeen six years ago, revealed that he has begun discussions over his future and hopes to be handed a new deal.
Mulgrew said, as quoted by the Scottish Sun: "I'm hoping to be back. I'm in discussions at the moment. It's kind of been put on the back burner until the new manager is sorted out, understandably.
"I've always said how much I love it here and hopefully we can get something sorted."
Though Tierney is now the established first-choice left-back, Emilio Izaguirre is set to leave the club this summer, and Mulgrew would be a dependable and experienced back-up for the 18-year-old.
Mulgrew's versatility may also be a saving grace as he looks for a new deal, as he is also able to play at centre-back and in midfield, a hugely valuable attribute as Celtic look ahead to another attempt at a prolonged European campaign next season.