The Scottish Premier League champions saw a bid for Espanyol defender Raul Rodriguez knocked back last night but director of football Gordon Smith believes there is good news afoot in terms of existing players.
While Juan Manuel Ortiz is the sole newcomer since Craig Whyte completed his takeover deal in May, the new regime have shown they can tie up a deal by securing Steven Whittaker and Allan McGregor on contracts totalling 11 years.
Davis is set to follow and Smith claims Bougherra could begin talks despite speculation over a summer exit.
"We have offered Steven Davis an extension to his contract and we're hoping we get that completed," Smith told BBC Scotland. "We are just waiting to finalise it."
He added: "Madjid Bougherra, we were always looking to have him this season playing for us.
"The possibility now is that he's interested in talking further with us with a view to extending his contract.
"A lot of people just thought that Madjid Bougherra said he didn't want to stay at Rangers.
"But that is certainly not the case and there is a possibility that he could be staying for a longer time."
Rangers could increase their offer for Rodriguez and they retain interest in Carlos Cuellar, Kenny Miller and David Goodwillie.
And 41-year-old defender David Weir is close to agreeing a deal to play on this term after appearing in the pre-season games in Germany.