Sources say the deal could be finalized within the next 48 hours.
Talks are centering around adding two more years to Westbrook's current contract, which expires after next season. The new deal would call for Westbrook's salary to jump from $17.8 million next season to the maximum $26.5 million. He would then earn the max the next two seasons.
Westbrook, who has played all eight seasons of his career in Oklahoma City, would have a player option for the third season of the deal that would allow him to capitalize on the league rule allowing players who have completed 10 seasons to earn a larger max salary.
While one source said the Thunder are optimistic about extending Westbrook, he cautioned that nothing has been finalized yet. The source added that Oklahoma City is not giving Westbrook a hard deadline by which he must agree to the extension.
Signing Westbrook would end a month of speculation about Oklahoma City trading the superstar this offseason after losing free agent Kevin Durant, who signed with Golden State, and getting nothing in return.
In the wake of Durant signing with the Warriors, the Thunder had made locking up Westbrook long-term their primary offseason objective.
Westbrook, 27, had a season for the ages in 2015-16, recording a league-high 18 triple-doubles — including six in the month of March — to tie Magic Johnson for the most in a single season in the past 50 years.
A five-time All-Star and two-time All-Star Game MVP, Westbrook averaged 23.5 points, 7.8 rebounds and 10.4 assists in 2015-16 and finished fourth in the MVP voting.
ESPN's Brian Windhorst and Royce Young contributed to this report.