Ribery, who turned 35 on 7 April, and 34-year-old Robben are in the final months of their existing deals, but Bayern are keen for the two long-standing club servants to stay on for another season.
"We've had discussions with Franck and Arjen in the last view days," Salihamidzic said after Bayern announced that Eintracht Frankfurt's Niko Kovac will succeed Jupp Heynckes as head coach in the summer. "They've both been offered contracts through to 30 June 2019. It's in their court - they just need to let us know. We'll know more in the coming days."
Ribery has showed no signs of slowing in his 11th season with the record German champions, becoming the fifth player in history to win eight Bundesliga titles after Bayern legends Philipp Lahm, Bastian Schweinsteiger, Oliver Kahn and Mehmet Scholl.
The Frenchman has made 381 competitive appearances for Bayern since moving to the Allianz Arena from Marseille in summer 2007, registering 117 goals and 178 assists. He is also the Bayern’s leading non-German appearance maker.
Watch: Ribery reacts to winning the Bundesliga for a record-equalling eighth time
Robben - who swapped Real Madrid for Bayern in 2009 - has been no less influential and enduring.
The seven-time Bundesliga winner has netted 138 goals and chipped in with 97 assists in 287 outings for the Bavarian giants to date, and famously converted Ribery's pass to score the winning goal in the 2013 UEFA Champions League final against Borussia Dortmund.
Watch: Arjen Robben's top 5 Bundesliga goals
The wing duo - nicknamed 'Robbery' - have also won multiple DFB Cups, the FIFA Club World Cup and the Supercup, among others.