The development comes after European Union leaders last week signed off on the first stage of Brexit negotiations, ending more than a year of stalemate over Britain's bill for leaving the bloc, as well the fate of the Irish border and European Union expatriates.
Ms Lagarde indicated she expects the EU's negotiators to do more to reach a deal, as well as the UK's team in Brussels.
Responding to Mr Barnier's suggestion that the kind of free trade agreement likely to be on offer to the United Kingdom will not include financial services, Mrs May said it was a matter for negotiation.
The Bank of England is reportedly set to unveil plans to allow European banks in Britain to continue to operate as normal after Brexit.
The EU has drawn fire from Britain over its insistence that in order to have the strategic rock included in a transition agreement Britain needs to discuss the issue directly with Spain and a document, due to be published on Wednesday, is expected to confirm Brussels' stance on the issue, according to The Guardian.
British Prime Minister Theresa May had asked for the transition to last around two years.
Mrs May told the Cabinet she was seeking "a significantly more ambitious deal than the EU's agreement with Canada", her spokesman said.
Britain will need a transition after Brexit that will avoid businesses having to adjust to new rules and regulations twice.
Earlier on Wednesday Christine Lagarde, the head of the International Monetary Fund, said that it was imperative that the United Kingdom secure a good deal for the City in order to minimise the economic impact of Brexit.
The next phase comes as Barnier ruffled feathers in London on Tuesday with a warning that any ties after Brexit would inevitably result in Britain's financial companies losing full rights to trade across the bloc.
They were still outnumbered by the 230,000 European Union citizens who arrived to live in Britain in the same period, although that figure was down 19 percent on the previous year.
May had been headed for a second defeat later on Wednesday after several members of her party said they would vote against writing the date of Britain's European Union exit, March 29, 2019, into law.
May has already accepted in theory that the ECJ will continue to have jurisdiction and that London will have no legal basis to conclude its own trade deals during transition the period.