Xabi Alonso has stepped up his comeback from injury, but has yet to decide whether to extend a Real Madrid contract which has entered its last 12 months.
Alonso, who underwent a groin operation on June 7, is continuing his recovery at Madrid's pre-season training camp in Los Angeles.
Early rumours that he would not return until October appear to have been wide of the mark, with coach Carlo Ancelotti saying the midfielder was close to playing again.
He is reportedly attracting interest from other clubs including Jose Mourinho's Chelsea, and officials at the Bernabeu are aware that talks on a free transfer could begin as early as January.
According to AS, the offer of a one-year extension was made to Alonso's agent in May, but no further talks have taken place.
The Spain international reportedly plans to see how he recovers from his injury - which severely restricted his contributions for club and country last season - before making any decision over his future.
In a question and answer session on Madrid's official website, he said he was enjoying the "intense" pre-season preparations as the Madrid squad got ready for the new campaign.
"I am adjusting slowly," he added. "We are getting in shape to start off the season as best we can."
When Alonso does return to full fitness, he will join a wealth of midfield options at the Bernabeu, with summer signings Casemiro, 21, and Asier Illarramendi, 23, adding to the competition.
He said he had been impressed with the new arrivals' displays in friendlies, adding: "The new signings are looking great. The more options the coach has, the better."
The Basque said it was important that Madrid, who kick off the new La Liga season at home to Real Betis on August 18, began well.
"We always start out with the maximum expectations," he said. "In a club like Real Madrid, you know that every match is vital, and we always want to come away with a win. There is a good feeling in the squad, now it is up to us to show that on the playing field."