"Whatever it takes, we will prevent a no-deal Brexit".
The president of the European Council Donald Tusk, who hosts next week's summit, had said he would have pulled the plug on plans to discuss Brexit if there was no hope of progress.
Addressing the Co-operative Party conference in Glasgow, Mr Starmer said that if Mr Johnson was unable to secure a deal at next week's crucial European Union summit, he must comply with the so-called Benn Act and seek a further delay.
But the details of Mr Johnson's concessions are not yet known.
"If Boris Johnson does manage to negotiate a deal then we will insist that it is put back to the people in a confirmatory vote", he said.
Johnson did not make a public comment after the meeting in an apparent attempt to rebuild trust between the leaders, which has soured in recent weeks, and avoid derailing any breakthroughs, but Downing Street echoed Varadkar's claim that the leaders had identified the "pathway to a deal".
Speaking to reporters Boris Johnson told reporters: 'I had a good conversation with the Irish Taoiseach Leo Varadkar yesterday and I think both of us can see a pathway to a deal, but that doesn't mean it's a done deal.
Negotiators have been given the green light for talks to enter the so-called "tunnel" phase - diplomatic jargon for talks continuing in secret between a small group of negotiators in complete secrecy - with a view to presenting a deal ahead of the European Union summit on 17 and 18 October.
Meanwhile, Mr Johnson will challenge MPs to back any deal he has secured from Brussels when parliament sits on a weekend next Saturday for the first time in nearly four decades, The Times has reported.
The Taoiseach said a deal was possible after more than two hours of talks but warned that "there's many a slip between cup and lip".
Mr Barnier is due to brief European Union ambassadors and MEPs on Monday on progress - but time is rapidly running out if they are to get an agreement in place in time for European Union leaders to sign off on it at next week's Brussels summit.
Mr Varadkar has reiterated time and again that there will be no hard border in Ireland- the offer of a deal from Mr Johnson last week was rejected on the grounds that there would be checks on goods-vehicles crossing between Northern Ireland and the Republic.
The EU strongly objected to a proposal in Mr Johnson's Brexit blueprint for Northern Ireland to leave the customs union together with the rest of the United Kingdom - meaning the return of customs controls on the island of Ireland. "You need vigilance, determination and patience", following his meeting with the UK's Brexit Secretary Stephen Barclay.
Varadkar's words Thursday indicated that Johnson might have shifted his red lines and could now be willing to keep Northern Ireland in the customs union.