The European Parliament's chief Brexit co-ordinator, Guy Verhoftstadt, said that another extension to the Brexit deadline would be "unacceptable unless the deadlock in London is broken."
"Let it be a second referendum, new elections, a revocation of Article 50 or the approval of the deal, but not today’s helpless status quo," Verhoftstadt said.
His comments came shortly after French Foreign Minister, Jean-Yves Le Drian, said his country would reject attempts to extend the Brexit process in the absence of a new approach from the British side.
"The British say they want to come up with alternative solutions for withdrawal and No Deal, we have not seen them, so it's 'no', we're not going to do it every three months," the minister said during an interview with Europe 1 on Sunday.
On Saturday, UK's Work and Pensions Secretary, Amber Rudd, announced that she was quitting Johnson's cabinet, saying that the government was not having any "formal negotiations" with the EU about a new deal and was simply preparing for a no-deal.
Chancellor of the Exchequer Sajid Javid responded to Rudd's claims, saying that Boris Johnson was "straining every sinew" to reach a new agreement with Brussels.
On 6 September, the House of Lords approved a measure that would block Johnson's plan to leave the EU without a deal. Previously, the bill was passed by the House of Commons and now proceeds to royal assent.
After the bill was passed, Mr Johnson wrote to Tory members saying: "They just passed a law that would force me to beg Brussels for an extension to the Brexit deadline. This is something I will never do."
Mr Johnson has repeatedly stated his determination to withdraw the United Kingdom from the bloc by the 31 October deadline, with or without a deal, despite there being a majority of lawmakers opposed to no-deal.
The United Kingdom has long been facing deadlock over its withdrawal from the European Union. A deal reached by London and Brussels last year was rejected three times by the House of Commons and eventually led to Theresa May resigning as prime minister.