The Swiss goalkeeper is in the final months of his contract at Tannadice.
According to the Daily Mail, the Scottish league leaders are interested in the 29-year-old with both of the club’s goalkeepers also out of contract at the end of the season.
Allan McGregor and understudy Jon McLaughlin are on deals which expire in the summer with little movement on that front. The former signed a new one-year deal in April and turns 40 at the end of the month.
Siegrist has attracted interest from England and Celtic over the past 12 months due to his form for United since helping them win promotion from the Championship.
Dundee United have begun preparing for life without the Swiss goalkeeper, signing Finnish international Carljohan Eriksson.
Siegrist is an attractive proposition to sides, especially those in Scotland where he has proven his quality over the past couple of seasons.
He is rightly regarded as one of the best goalkeepers in Scotland. A dominant presence in the box, agile, confident with the ball at his feet and capable of making spectacular saves.
So far this campaign, going by Wyscout metric 'prevented goals’, Siegrist out performs both McGregor and McLaughlin, as he did last season when he led the league for the number of goals prevented at more than 12.
For Rangers, Siegrist would be a low risk signing with potentially two goalkeepers needing to be replaced. He has proven his reliability at Premiership level.