Mr Street said it is crucial he is given a seat at the table at Cobra or another decision-making committee, as each region will have different priorities when it comes to easing the lockdown.
The Mayor said he understood that decisions up to now had been taken by politicians based in London as enforcing the lockdown and social distancing measures were done in the national interest at a time of emergency.
But he added it was now time for him to be invited into the heart of Government discussions. London Mayor Sadiq Khan and Scottish First Minister Nicola Sturgeon attend meetings of Cobra, the Government's emergency committee, but Mr Street and Greater Manchester Mayor Andy Burnham do not.
The former boss of John Lewis said he would be keen to see schools reopen and the housebuilding sector start up again when it is safe to do so.
He said: "If we are moving into the recovery stage, which will hopefully come fairly soon, it's really important the West Midlands has a clear voice in that. I want to be right at the table.
"I don't know if it will be Cobra or not, it could be a sub-committee of Cabinet, it could be individual ministers but I want to be absolutely involved in discussions on recovery plans. Every region is slightly different."
On easing the lockdown, the Mayor said: "I don't want a difference between London, the West Midlands and Manchester. I think it's extremely difficult to do."
He added the time would soon come to think about a phased end to the lockdown, following reports the Government is considering a traffic-light system, possibly starting with re-opening schools and non-food shops.
Mr Street said: "I'm keen to see schools open for key workers. I'm keen to see construction getting back to full speed, particularly the construction housing sector. It employs a lot of people and can be done with sensible social distancing."
The Tory Mayor defended the Government over its response to the coronavirus outbreak, following a damning Sunday Times report in which revelations included Boris Johnson missing five Cobra meetings, but acknowledged the region's hospitals had not got the levels of personal protective equipment (PPE) needed.
He said: "There is huge precedent for the Prime Minister not chairing Cobra and a department minister doing it, so I wasn't hugely taken by that.
"I would accept the situation on PPE and the West Midlands not getting what it expected to get. It is true they have not received PPE they have ordered and were expecting to have.
"It is not yet solved and everybody is working to get a solution to it."