The government has said it hopes to roll over around 40 EU free-trade agreements with third countries by the time Britain leaves the bloc.
It has said it has reached continuity agreements with Switzerland and Israel as well as mutual recognition agreements with Australia and New Zealand, but Hollingbery said the government had not yet formally signed any free trade deals.
“I believe that the majority of those will be in place by March 29,” he told parliament.
“Some of them are challenging, one or two are even more than challenging - they are close to impossible,” he added, citing Turkey as one such difficult country as it is part of a customs union with the European Union.
Trade minister Liam Fox has said there is “a pipeline of agreements” to be signed in the coming weeks and is meeting his counterparts from around the world at the World Economic Forum in Davos to push them to agree to replicate the deals.
May’s exit deal with Brussels was rejected by a large majority in parliament last week, and MPs are due to debate the way forward on Jan. 29.
Hollingbery said that if a deal were reached by March 29, the “time pressure” would be reduced on rolling over trade agreements because of a transition period with the EU due to last until December 2020.