The 30-year-old signed for Manchester United on loan from Crystal Palace in the January transfer window.
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However, he is yet to make an appearance for the Premier League giants, having been named on the bench 11 times.
Butland had initially been linked with Rangers last year, but he could be set to make the move in the coming months.
As reported by The Scottish Sun, he is 'on the brink' of becoming the Gers' first signing of the summer window.
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Boss Michael Beale is said to be seeking a replacement for 41-year-old Allan McGregor, who is set to leave at the end of the season.
Beale is also an admirer of Butland, whose contract at Palace will expire at the end of June.
The goalkeeper will be given the no.1 shirt when he completes his move to Ibrox.
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Butland is yet to make a competitive appearance this season, with a fractured finger keeping him sidelined for the first half of the campaign at Palace.
He has previously featured nine times for the England national team and has made a total of 293 club appearances.
United are the eighth club and seventh loan of his career, which started at Birmingham City before he moved to Stoke City and then Palace.
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Butland was an unused substitue in the Red Devils' FA Cup semi-final win against Brighton on Sunday.
David de Gea remains as Erik ten Hag's first choice option in goal, although his deal at the club is set to expire at the end of the season.