The Labour leader told MPs they "must now take control".
She replied: "I will be voting for the motion standing in my name".
The BBC's political editor Laura Kuenssberg said the prime minister was "likely" to head to Strasbourg later - where the European Parliament is based.
Conservative MP John Baron then pushed Mrs May to back a no-deal Brexit, arguing "no deal is better than a bad deal". "That's why MPs must reject this deal today", the politician noted.
The MP voted with the Prime Minister but earlier in the day warned Brexiteer MPs, who favour a "no deal" Brexit. And Corbyn's plan includes a new customs union with the EU to solve the issue of the border in Ireland.More news: Elon Musk's SpaceX capsule splashes down off Florida coast
Her exchanges with Corbyn involved the Labour leader repeatedly urging the prime minister to outline a new plan for Brexit, and to consider backing the Labour proposal, under which the United Kingdom would stay in some form of customs union.
"They want to leave the European Union, end free movement, have our own trade policy, ensure laws are made in this country and judged in our courts".
In a statement, the Commission said it had put forward proposals to try and reassure MPs the backstop "if used will apply temporarily".
Mr Corbyn, in his reply, said: "I do have sympathy with the Prime Minister on her voice and I hope it soon recovers, I understand how painful this is".
"The prime minister's deal has failed, she no longer has the ability to lead", he said. "The British public were asked by Parliament if they wanted to leave, they said they wanted to leave, so Parliament needs to show that it will deliver that".
Should MPs reject that, there will be another vote on whether Parliament wants to seek an extension to Article 50 - delaying the UK's departure beyond the current March 29 deadline.More news: Venus Williams makes spirited comeback as Keys stumbles at Indian Wells
"He has nothing to offer this country".
He said: "Prime Minister, have you had the opportunity to consider whether you would support that amendment?"
Ms Allen said she would need to look carefully at the wording of the amendments that are brought forward tomorrow and Thursday - already there is a suspicion that the government might be "playing games" with the suggestion that "one is linked to the other".
She has also said she would be voting against the potential for a "no deal" scenario at another crucial vote on European Union membership in The House of Commons tonight, March 13.
MPs who voted for: Sir Peter Bottomley (Worthing West), Ms Caulfield, Nus Ghani (Wealden), Nick Gibb (Bognor and Littlehampton), Nick Herbert (Arundel and South Downs), Gillian Keegan (Chichester), Stephen Lloyd (Eastbourne), Mr Loughton, Huw Merriman (Bexhill and Battle), Jeremy Quin (Horsham), Amber Rudd (Hastings and Rye), Sir Nicholas Soames (Mid Sussex).
The prime minister added: "We can not serve our country by overturning a democratic decision of the British people".More news: Conor McGregor arrested in Miami on robbery, criminal mischief charges