Chancellor Rachel Reeves has met with European finance minsters to discuss the future as the UK government considers seeking a wide "reset" of post-Brexit EU relations.
While Labour says rejoining the trade bloc is not on the table, the lifting of barriers to farms and food exporters could be an option, the BBC reports.
Help could also be provided to firms facing complex supply chains and European leaders want a focus on boosting youth mobility.
Negotiations would begin in proper in the new year.
After giving a speech to Eurogroup finance ministers in Brussels - the first given by a UK chancellor since Brexit - Reeves emphasised that the UK would not rejoin the single market and customs union or Freedom of Movement.
But she did not rule out keeping UK laws in line with EU regulations for farm, food and goods exports, a policy known as "dynamic alignment".
The UK has previously sought an agreement that could allow meat, fish, dairy, and live animals less stringent checks when crossing a UK-EU border.
Extending the Brexit fisheries deal beyond 2026 is likely to be a condition for any wider reset.
Criticising the developments, the Conservative Party said it "looks like Labour's plan is to follow EU rules for nothing in return".