The team remain undecided whether to keep the Japanese driver in a race seat for next season.
Team boss Monisha Kaltenborn told Sky Sports News: "We've not made a decision on the second seat as yet.
"We know [Kobayashi] well and we know his qualities. But we will announce during the season - and it's not that long any more."
Kobayashi claimed his first F1 podium earlier this year at his home race, the Japanese Grand Prix.