The Canaries are understood to be keen on hanging onto the 30-year-old former Wolfsburg centre-back, but The Sun reports that the Ibrox side are also keen on the 16-cap international.
Klose’s Carrow Road deal is up in June, but discussions have taken place between the newly-promoted side and the defender about a new deal.
Norwich’s promotion to the English Premier League leaves them in a position to be able to offer more money if they are keen on holding onto Klose.
Rangers will likely be looking for central defensive reinforcements, with doubts over the Ibrox futures of on-loan Nottingham Forest defender Joe Worrall and out-of-contract Northern Ireland veteran Gareth McAuley.
German-born Klose has played 33 times for the Canaries this season, and has played more than 100 games since his £7.6 million move from the Bundesliga in the January 2016 transfer window.
Meanwhile Borna Barisic, Eros Grezda and Kyle Lafferty could all find themselves told to move on in the summer, amid reports the trio incurred Steven Gerrard’s wrath with their performances in the 4-4 draw with Liverpool Under-23s.
Gerrard’s displeasure at some of his first-team stars in the closed-doors game was made clear, the Light Blues boss saying in the aftermath: “It was a very useful game. I probably learned more in that 90 minutes about the players than I have done all season. First half was an embarrassment and the second half was fantastic. It was a very useful exercise for different reasons but in terms of us as staff, we learnt so much about the players invididually.”