Speaking ahead of a virtual question-and-answer "Call Keir" session with members of the public from Glasgow, Sir Keir appeared to back Nicola Sturgeon's approach to keeping lockdown in place for another three weeks.
Boris Johnson has said that he will make an announcement on Sunday about lifting restrictions, with some kicking in on Monday.
Asked if he trusted Boris Johnson or Nicola Sturgeon more on when restrictions should be lifted, he said: "It's not a question of who I trust more. It's really important in Scotland, and across the UK, that we're cautious about this and nothing is done which might send the infection rate up above one.
He added: "The risk of not doing everything a the same time means different rules for different places which makes it much more difficult to police. If the restrictions are lifted in one area which meant you could travel across borders, then it's very hard to see how you could have seperate regimes in countries or regions. We should go cautiously, putting the protection of lives as the priority, and if at all possible it should be a four nations approach.
"It's very important we build a national consensus of what happens next and that means the Prime Minister has to involve the devolved administrations, the trade unions and civic society and opposition parties in a very meaningful way."
Sir Keir said he had spoken with Boris Johnson earlier in the day and that he had "made it clear from our point of view that there shouldn't be any easing of restrictions until we're sure the infection rate is under control. There are many people who haven't had the infection and therefore if the R number [transmission number] goes above one there's a real risk of a second phase and everyone wants to avoid that."
He also raised concerns about a potential financial "cliff-edge" for hundreds of thousands of people if the furlough scheme is ended next June, after Chancellor Rishi Sunak suggested it be wound up quickly.
"Where some businesses go back to work sooner than others they need flexibility so it's not all or nothing," he said. "And we can't have a cliff-edge at the end of June when the furlough scheme runs out.
"Furloughing ends in June and if it's not extended, from next week businesses will have to start redundancy processes. UBI is a medium to long term issue, the practical issue in the next week is are those furlough schemes going to be extended, because if they're not a lot of employers are going to have to start statutory consultation exercises. The danger there is the good work done by the furlough so far in keeping people in their jobs will be lost.
"UBI isn't something we have supported that and there's a healthy debate going on, but I'm focused on the furlough scheme at the moement."
He also said that he would not exploit the current crisis as a way of attracting voters back to Labour. "I don't think there's any sense the Labour Party, or any political party, should seek to gain advantage from this crisis. We should all pull together in the national interest. A crisis doesn't give the Labour Party any opportunity at all."
But he added: "I'm deeply conscious my job is to rebuild trust in the Labour Party, particularly in Scotland, which is why I very much wanted to be in Scotland and instead we're hosting these virtual sessions. I know that re-engaging across Scotland with different communities in what will be pretty frank discussions I hope, is the way to rebuild trust. I will be listening to what people say. But we've a job of work to do, crisis or no crisis."
Mr Starmer aslso said he believed the disciplinary case of the suspended nine Aberdeen councillors had gone on "far too long" and needed to be resolved, and pledged to make changes to procedures to ensure cases were dealt with more quickly.
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