The shadow chancellor remarks come as Mr Johnson embarks on a charm offensive over the next 24 hours, frantically lobbying opposition MPs to support his deal after he failed to secure the DUP's backing.
Labour has ordered its MPs to reject the prime minister's deal during Saturday's session of the House of Commons – the first weekend sitting since the Falklands conflict.
But Mr McDonnell declined to say whether Labour rebels would lose the party whip if they voted with the government, adding: "This is a three line-line whip and the chief whip, in the normal way, will determine the consequences for anyone who doesn't vote for it."
"I"m not the chief whip thank goodness; I've got enough on my plate," he told BBC Radio 4's Today programme.
So far, only a handful of Labour MPs have publicly suggested they will back Mr Johnson's deal that EU leaders signed off on Thursday at the European Council summit.
Speaking on BBC Newsnight, the Blyth Valley Labour MP Ronnie Campbell said he intended to vote for the deal, adding: "It's time this country this country just got rid of the European issue and got down to the arguments that are on the streets."
Asked about his remarks, Mr McDonnell said he would "have a chat" with his colleague, claiming: "He's a good socialist and all his life he has fought alongside me to protect trade union rights and developing trade union rights."
"On this one I'm going to have a chat with him and point out to him: please do not give him this power (to weaken workers' rights) to Boris Johnson because you know what he'll do.
He continued: "He'll undermine trade union rights. Boris Johnson and those Tory MPs that populate the cabinet, these are the extreme right who have attacked trade unions throughout their political career.
"No MP, as far as I'm concerned, who has the true interest of their constituents at heart can allow that to happen."
But Graham Stringer, a second Labour MP, who voted against Theresa May's Brexit deal, said on Friday he could also vote for the new agreement, adding he will "consider whether, if this deal goes down, we won't get Brexit at all."
He added: "But if I thought that we wouldn't get Brexit at all, then I would consider voting for it."
Reiterating the DUP's opposition to the prime minister's Brexit deal, Sammy Wilson said: "I can give you an absolute assurance we will not be voting for this deal when it comes to the Commons tomorrow.
He said voting for the Withdrawal Agreement would support "siphoning us off from the rest of the UK" and argued it would mean businesses in Northern Ireland incurring "additional costs and administrative burdens".
Mr Wilson said the DUP made concessions with Mr Johnson in order to help him get a deal but suspected he would do what was best for the Conservative Party. "We are disappointed he didn't stick to the red lines he said he would," the DUP MP added.
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