"We need a proper contest with the best candidates on the field, and I will be standing," Streeting told a conference of the Progress group of Labour supporters, who view themselves as being on the modernising wing of the party.
Addressing the group, Streeting described the United Kingdom's 2016 referendum decision to leave the European Union as "a catastrophic mistake" that had made the country its weakest since before the Industrial Revolution.
He said the UK should seek to rejoin the bloc.
"We need a new special relationship with the EU, because Britain's future lies with Europe, and one day - one day - back in the European Union," he said.
Starmer opposed the UK leaving the EU but as prime minister has rejected trying to rejoin or to make major concessions on areas like immigration, which the EU sees as essential for much closer economic ties.
Late on Thursday, Greater Manchester mayor Andy Burnham said he was standing in a by-election in a newly vacant seat to return to parliament, a precondition for him to be able to challenge Starmer as prime minister.
Starmer has rejected calls to step down after his party suffered a heavy defeat in local elections across England last week.
Labour party rules require each challenger to get support from at least 20 per cent of the party's members of parliament before being able to trigger a leadership election.
