He pledged to give parliament a fresh vote on Britain's options by February 27 if May does not come back with new concessions before then.
He said he believed the prime minister was "pretending to make progress" but actually meant to return to Parliament after the March 21/22 European Council summit the week before Brexit and offer MPs a "binary choice" - her deal or no deal.
"I'm certain of one thing, is that it's not going to be as good as if they had not been Brexit, that is for sure", Lagarde said.
May held talks in Brussels last week in an attempt to secure changes to the Northern Ireland backstop as demanded by Tory MPs.
Labour will seek to force Mrs May into a decisive second Commons showdown on her Brexit deal by February 26.
Britain has reached an impasse in its last-minute renegotiation of an exit deal that it agreed with the European Union a year ago but that was overwhelmingly rejected by the British parliament in January.
"I think that gives that sense of timetable, clarity and goal on what we are doing with the European Union - taking that work forward and our determination to get a deal - but equally knowing that role that Parliament very firmly has". While UK PM May's comment might appear to sooth an ongoing drama over Brussels, the European Union leaders had asked for insurance for Northern Irish borders, which appeared to be complicating the overall complexion of the recent round of talks.
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Others threw projectiles at police, who responded with tear gas and stun grenades to disperse crowds. But France's Interior Ministry said this week's protest was significantly smaller than last week's.
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TRD Off-Road is also available in Midnight Black or Lunar Rock, both of which are available with optional Ice Edge Roof. The new Sequoia TRD Pro offers plenty of storage inside and out for 2020 with room for seven passengers.
There are fewer than 50 days until Brexit.
"Not only will this help to support jobs throughout the United Kingdom but it will also be a solid foundation for us to build an even stronger trading relationship with Switzerland as we leave the EU", International Trade minister Liam Fox said in a statement.
But it was rejected by the UK Parliament and if it is not approved by Brexit day, the default position would be a no-deal Brexit.
Last week, Labour leader Jeremy Corbyn set out the conditions under which he would instruct his party to support an exit deal in parliament.
A number of government ministers will also be meeting their counterparts across the continent this week, in order to underline Mrs May's determination to achieve a deal.
The EU and the Irish Government have insisted that the backstop is the best way of guaranteeing there will be no hard border in Ireland.
The UK's exit from the bloc may lead to a return of a hard border between the Republic of Ireland and Northern Ireland in potential violation of the 1998 Good Friday Agreement, which specifically stipulates that no physical border should exist between the two sides.