Mr Street said the region may need to go into tier three "very high" alert measures if Covid cases continue to surge, meaning pubs would be forced to shut and people from different homes banned from mixing.
But he said if ministers proposed such a move, it would only be acceptable if an economic support package was agreed in advance.
Sandwell, Walsall, Wolverhampton, Birmingham and Solihull are currently under tier two "high" alert restrictions, while Dudley and Coventry are in tier one.
The measures are due to be reviewed next week.
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Mayors in the north of England, including Greater Manchester's Labour Mayor Andy Burnham, have been in a stand-off with ministers after rejecting tier three proposals.
Conservative Mr Street refused to be drawn on whether they should accept the tougher measures, but said: "If we suddenly found ourselves in a position where we had to move into higher restrictions, first of all I would want to do it with all of the local authority leaders, and if we were at that high level of infection, I think it would be very important that we do not deny the health reality.
"I would not want to stand in the way of proper health restrictions.
"However, I would want the economic support to be agreed in advance. I just do not believe that it is right that the negotiation is going on in the public way that it is.
"That should be a pre-ordained package. We should be clear about what our position would be well before we get to that point."
Mr Street has been backed by Walsall Council leader Mike Bird and Birmingham Council leader Ian Ward.
Mr Bird urged Ministers to set out the criteria "to get out of tier two going downwards".
He added: "I hope that we don't see any of us going into tier three, but in actual fact the measures that we have put in place should see us going down the ladder rather than up the ladder."
Mr Ward said: "Keeping people safe has to be our number one priority, but we also expect an economic support package."
Tier three restrictions currently apply in Liverpool and parts of Lancashire, where infection rates are above 600 cases per 100,000 people.
Birmingham has the highest rate in the West Midlands at 166.2 cases per 100,000 people.