Lord Walker is the latest politician to suggest the fuel duty cut should remain past September.
Some Labour MPs have privately questioned why Sir Keir has not committed to ditching the increase, and opposition parties have called for further cost-of-living support to offset the spike in oil prices.
The Conservatives, Reform UK and the Lib Dems have all argued that the war in Iran means the chancellor should keep the cut in place.
Conservative leader Kemi Badenoch has said she is predicting Sir Keir will be forced to announce “another humiliating U-turn”. She has also called for cuts to taxes on energy bills.
Instead of waiting until September, Liberal Democrat leader Sir Ed Davey said the government should introduce an emergency 10p cut to fuel duty immediately.
Reform’s Treasury spokesman Robert Jenrick has also urged the government to halve VAT on petrol for three months.
The Green Party of England and Wales said the government should make plans to cover potential energy bill rises of up to £300 per household.
The SNP wants energy policy to be devolved to the Scottish Parliament in Holyrood, and has argued that Scottish independence would help reduce bills.
Plaid Cymru has said long-term investment in renewable energy would help protect people from price hikes.