Two-time world champion Alonso signed a two-year deal with the option for a 12-month extension when he made the shock decision to re-join the Woking-based outfit in 2015, patching up his differences with McLaren after his acrimonious departure in 2007.
Having returned in the hope of winning a third world title, the 35-year-old has found the going tough, finishing 17th last season in the standings, while he improved to end the 2016 campaign in 10th spot.
Alonso has said he will wait until the end of the 2017 campaign, when his contract is due to expire, to decide whether he wants to continue driving for McLaren, admitting he wants to see what kind of car they can produce under the new rules.
Capito hopes McLaren can persuade Alonso to stick around, claiming he has formed a good relationship with the Spaniard during his three months in his role as CEO.
“If the car and the engine are good, and he enjoys the new rules, there is no reason why he shouldn’t stay,” Capito told BBC Sport. “Fernando is a great asset for the team and we should do everything to keep him.”
Southampton have announced plans for a two-week tour of China as part of their pre-season preparations for the 2018-19 season.