Shadow health Secretary Monica Lennon said the Scottish Parliament must be allowed to decide on the issue, while defeated Rutherglen and Hamilton West MP Ged Killen said the SNP now has a mandate for a vote on leaving the UK after their landslide election victory north of the border last week.
It will increase the pressure on Richard Leonard to change course on the issue after insisting that the 2021 Holyrood elections will be test of a mandate for a second vote on leaving the UK.
Ms Lennon has admitted that "many Labour voters" were among those who backed the SNP last week.
“As expected, Nicola Sturgeon is presenting that as an endorsement of her party and will now ask the UK Government to permit a second referendum on independence," she told the daily Record.
“If Boris Johnson isn’t prepared to grant this request, he should allow the Scottish Parliament to decide.
“The SNP blueprint for independence is flawed and will disappoint many progressive Scots who are fed up with austerity. Nevertheless, the future of Scotland must be decided by the people of Scotland."
Mr Killen who lost out to the SNP's Margaret Ferrier the Lanarkshire seat also backed calls for a Labour rethink.
"I campaigned on a promise to vote against indyref2, but I lost," he said.
"The SNP made massive gains on a promise to hold another referendum and, as democrats, we must accept it even if we don't like it."
Malcolm Chisholm, a former Labour minister at both Holyrood and Westminster also said the party had to change.
“Hope many of those against or undecided (me included) about independence will get behind incontrovertible democratic demand for IndyRef2," teh former Edinburgh North and Leith MP added.
“Hardest of hard Brexits and intolerable Johnson not what many No Voters had in mind in 2014.”
Lothians MSP Neil Findlay, a close ally of Jeremy Corbyn, said "tens of thousands" of Labour voters backed the SNP in the election.
"We cannot deny the people of Scotland a second referendum where the majority is calling for it," he added.
But he said a "clear proposition" would be needed from the SNP and this can't happen n 2019 because the outcome of Brexit won't be known.
It came after one of Labour's most influential councillors in Scotland, COSLA president Alison Evison also came out for an support of an in independence referendum at the weekend.
The Labour group leader on Aberdeenshire Council said democracy had become fragile in Scotland and needed to be "strengthened."
“We can strengthen it by enabling the voice of Scotland to be heard through its formal processes and that must mean a referendum on independence,” she added.
SNP Westminster leader Ian Blackford welcomed the shift in opinion from the Labour figures.
“The SNP have a cast iron mandate to hold an independence referendum and the UK government must respect that.
“It is welcome that senior figures within Labour are recognising that mandate.
“While we will not all agree on independence, we can surely find common ground that this decision must be in Scotland’s hands - not Boris Johnson's.”