Doucoure scored seven goals in a tremendous season for the Hornets as he put himself firmly on the radar of many bigger clubs in the Premier League.
He made a habit of scoring in games against high profile opposition, netting against Liverpool, Chelsea and Manchester United.
Arsenal are yet to appoint a new manager as a successor to Arsene Wenger, but Nicholas urged the next man in charge to go in for the French midfielder.
“It has to be Abdoulaye Doucoure by a distance,” Nicholas told Sky Sports when asked who Watford's player of the season was. “Even watching them lose to Spurs last Monday, his energy and movement stood out.
“I’m not a big holding midfield fan but he could easily do that and more for Arsenal with the way he covers the ground. Everton will surely be looking to snap him up as well.”
Arsenal and Everton are both looking for midfield reinforcements after disappointing Premier League campaigns.
The Gunners finished 6th in the Premier League and were eliminated in the Europa League semi finals, forcing the club to part company with Wenger after 22 years. Mikel Arteta is in line to take over at the Emirates Stadium.
Everton finished 8th but were inconsistent in their results and agricultural in their style of football. Their manager Sam Allardyce was sacked on Wednesday, with Marco Silva favourite to succeed him.
Doucoure is contracted to Watford until 2020, having signed from Rennes in 2016.