The 26-year-old joined the Bhoys from Scottish Premiership rival Dundee during the January transfer window, and made a strong impression.
He is out of contract with his parent club in the summer which means Celtic can sign him on a free transfer. However, the Bhoys could face strong competition for his signature, as Championship outfit Wolves and Premier League club Stoke City are reportedly keen to sign him as well.
Bain has impressed during his spell as a stand-in for Craig Gordon, and has expressed his desire to continue with the Glasgow club beyond the summer.
Brendan Rodgers has sung glowing praises of his ability in recent weeks, and Celtic are poised to make an official move to secure a permanent deal for him.
The report claims that the Celtic hierarchy will prepare to begin negotiations with Bain’s advisors shortly.
Meanwhile, the Bhoys are on the verge of winning their seventh consecutive Scottish Premiership title, and their second under Brendan Rodgers, after they won 3-0 against Ross County on Saturday.