The London mayor accused the Tory pair of stoking "the politics of division".
The Sky presenter said voters did not know what the party stands for any more.
The former Tory deputy chairman lost the party whip after failing to apologise for his attack on Sadiq Khan.
The deputy prime minister said his former cabinet colleague did not need to apologise.
Trevor Phillips poked fun at the GB News anchor after he lost the Tory whip.
The former Match of the Day host has backed a campaign criticising the government's Rwanda policy.
Trevor Phillips hit out after the prime minister said the UK could be "overwhelmed" by foreigners.
The deputy prime minister was left squirming on Sky News when shown a list of Tory failures.
The deputy prime minister suggested UK "civic society" has treated Israeli deaths differently to George Floyd's murder.
The deputy prime minister refused to say whether he had read all of the government's plans for dealing with a pandemic.