Boris Johnson has said before mid-February he will look at what lockdown restrictions in England could be lifted.
But he warned against any “premature” easing of the rules which could lead to “another big surge in infection”.
The government has promised to offer the vaccine to around 15 million of the most vulnerable people, including everyone over 70, by February 15.
The prime minister said on Monday before that date he would be “looking at the potential of relaxing some measures”.
It is understood no rules will be lifted before February 15. The legislation enforcing the current lockdown in England does not expire until the end of March.
Johnson is under pressure from some Tory MPs to set out a roadmap for how and when restrictions will be lifted.
But speaking to reporters on Monday, the prime minister would not guarantee schools would be back before Easter.
“Believe me there’s nothing I want to do more than reopen schools, I’ve fought to keep schools open for as long as I possibly could,” he said.
“We want to see schools back as fast as possible, we want to do that in a way that is consistent with fighting the epidemic and keeping the infection rate down.”
Having raised the prospect of lifting some restrictions, Johnson also confirmed he was “definitely looking at” the possibility of stricter rules for travel.
Ministers are set to meet on Tuesday to discuss requiring travellers arriving in the UK to quarantine in hotels to ensure they are following the rules on self-isolating.
“We have to realise there is at least the theoretical risk of a new variant that is a vaccine-busting variant coming in, we’ve got to be able to keep that under control,” Johnson said.
He added: “Daily we’re looking at the data and trying to work out when we’re going to be able to lift restrictions.
“Schools obviously will be a priority but I don’t think anybody would want to see the restrictions lifted so quickly while the rate of infection is still very high so as to lead to another great spread of infection.
“We’ve now got the R down below 1 across the whole of the country, that’s a great achievement, we don’t want to see a huge surge of infection just when we’ve got the vaccination programme going so well and people working so hard.”
The latest government figures showed more than 6.3 million people across the UK have received their first dose of the vaccine. On Saturday a record-breaking 491,970 jabs were injected.