The parties are divided over the Northern Ireland Protocol with unionists opposed due to concerns that additional trade checks have placed a border in the Irish Sea.
Sir Keir and shadow secretary of state Louise Haigh met the five largest parties at Stormont on Thursday during two days of official engagements in the region.
DUP leader Sir Jeffrey Donaldson said he urged Sir Keir to work with his party to “restore Northern Ireland fully within the UK internal market”.
Under the terms of the Protocol, Northern Ireland effectively remains part of the European single market and therefore is obliged to carry out additional checks on goods arriving into the region from Great Britain.
Unionists argued that this undermined the constitutional status of the UK.
There have been demonstrations and legal actions by loyalists and unionists against the Protocol.
Sir Keir said some kind of veterinary agreement between the UK and EU could minimise the checks, and urged flexibility on both sides
Sir Jeffrey said that through his previous work in Northern Ireland, Sir Keir will know “the need to maintain the delicate balance which has secured progress here”.
He argued the Protocol “fundamentally” damages the principle of consent and three-stranded approach espoused in the Belfast and St Andrew’s Agreements.
“The Protocol undermines those foundational agreements and the consensus so necessary for progress,” he said.
“I underscored my commitment to making Northern Ireland work and urged Sir Keir and his team to work with us as we seek to deal with the flawed Withdrawal Agreement and restore Northern Ireland fully within the UK internal market.”
Deputy First Minister and Sinn Fein vice president Michelle O’Neill said she shared Sir Keir’s view around a veterinary agreement as “the way forward”.
“There was a lot of common ground there in relation to the Protocol, we had a good conversation with him today,” she said.
SDLP leader Colum Eastwood said Prime Minister Boris Johnson “has to get his head around that you can’t have a hard Brexit and also avoid checks in the Irish Sea”.
“He can’t pretend that he didn’t understand what was in the Protocol that he negotiated, that he signed, that Lord Frost negotiated. They can’t pretend that they don’t know what was in that,” he said.
“They now have to face the reality.”
Alliance deputy leader Stephen Farry also said he believes a veterinary process “would go a long way to addressing many of the checks across the Irish Sea and defusing tensions”.
“It’s also important to highlight the continued manipulation of the situation around the Protocol by the UK Government and the ramping up of tensions on the Protocol,” he said.
“The continual whipping up of fears and the refusal to actually acknowledge what they themselves negotiated and the need for pragmatic solutions with the European Union, both of us remain extremely frustrated at the approach that the Government is taking.
“We do not see it as being serious and indeed respectful of the situation here in Northern Ireland.”