Labour MPs are trying to depose Jeremy Corbyn with a motion of no confidence
"I think the European referendum was a test of leadership and Jeremy failed that test," Hodge told Sky News.
Hodge added that, if the motion of no confidence passes, Corbyn "should do the decent thing and the same thing David Cameron has done."
A motion of no confidence does not mean that Corbyn would immediately have to resign.
However, it would add to pressure on the Labour leader, who has faced criticism for lacklustre campaigning during the EU referendum.
The letter, sent to John Cryer, chairman of the Parliamentary Labour Party (PLP) will result in a discussion about Corbyn's leadership at the next PLP meeting on Monday.
It could then lead to a secret ballot of MPs on Tuesday.
Fifty-five Labour MPs have signed the letter urging the left winger to quit, according to PoliticsHome.
Angela Smith MP earlier called for Corbyn to quit. Smith said: "Jeremy Corbyn has got to take responsibility. He should consider his position. He's shown insufficient leadership."
However, Labour MP Dianne Abbott leapt to Corbyn's defence this morning.
In a piece for the Guardian, the Shadow Cabinet minister wrote: "Jeremy Corbyn united the Labour party under his positive position of remain and reform, but this nuanced position, the only honest one of the campaigns, was ignored wholesale by a media more interested in the drama of Tory splits."