David Moyes’ side will be without the injured Ryan Fredericks until at least after the international break, with Ben Johnson their only available option at right-back at the moment.
West Ham lost Jeremey Ngakia to Watford at the end of last season while Pablo Zabaleta retired, and are keen to strengthen at the position and are considering a bid for Tavernier.
The 28-year-old has also been linked with Burnley and Brighton after he ended last season with 15 assists and three goals, but it’s the Hammers who are set to make an offer.
Tavernier already has six goals and three assists in 11 appearances this term, and his impressive form in the final third is attracting West Ham who could do with more goals from every area of the pitch.
He has two years left on his deal, and the Ibrox outfit are looking to renew his contract.
Tavernier wasn’t able to prove himself in the English top-flight while at Newcastle United, and has always been linked with a return.
It remains to be seen whether he will be keen to move to West Ham in the coming days, but the Hammers will have to table a very juicy bid to have the slightest chance of convincing him and Rangers.