Moyes is out of contract at the end of the season and talks between the two parties have slowed down amid a poor run of form.
The Hammers are without a win in five games and are currently on a three-game losing streak in the Premier League.
Many West Ham fans do not want Moyes to get a new contract and are in favour of bringing in a more attacking manager in the summer.
However, Warburton stressed that West Ham fans should be careful about what they want as the Scot has made them a regular feature in Europe and they are now a top-ten side in the Premier League.
He indicated their desire to change the manager might not bring the results they are wishing for.
Warburton said on the No Tippy Tappy Football podcast: “They are consistently in Europe, it’s their third year.
“They have won a trophy, they are in the Europa League, they are in the top ten so be careful what you wish for West Ham fans.
“I have got loads of mates who are West Ham fans in the city and they come out, moaning, ‘We only won 1-0, we were shocking’.
“It’s three points, you are sixth [ninth] in the table and you are in Europe; you have got to be really careful about what you desire.”
West Ham will look to get back to winning ways when they host Brentford at the London Stadium on Monday night.
