Hazard, 27, was the match-winner as the Blues beat Manchester United 1-0 in the FA Cup final at Wembley on Saturday.
However, a fifth-placed finish in the Premier League means no Champions League football next season and Clarke told BBC Radio 5 live’s Sportsweek the club must “rebuild” over the summer.
Goalkeeper Thibaut Courtois has also urged the board to invest in the squad.
“We’ll see after the World Cup whether I am a Chelsea player for next year,” said the 26-year-old Belgium international, who is out of contract at Stamford Bridge next summer.
“You see how Manchester City and Manchester United are spending and working.
“If we want to aim for the top, we should do the same – obviously within the limits of what’s possible.
“We’ll see what’s going on and I am sure the board will do what needs to be done.”
Chelsea lifted the FA Cup for the eighth time on Saturday thanks to Belgium forward Hazard’s first-half penalty, which he had won by being fouled by Phil Jones.
Hazard, who has two years left on his contract, joined Chelsea from Lille in 2012 and won the Professional Footballers’ Association Player of the Year award as they won the title in 2015.
But Clarke, a former Chelsea defender and assistant manager between 2004 and 2008, thinks it is “possible” the Belgian will leave Stamford Bridge this summer.
“You can always look to replace players,” said Clarke, now manager of Scottish Premiership side Kilmarnock.
“They come and go, they always move on. The club is the most important thing.
“They have to rebuild that squad a little bit. It wasn’t good enough to be in the top four. They have got a lot of work to do over the summer.
“To help fund other transfers for the club this summer maybe they do have to sell Hazard.”