Vestea, 52, is the council president of the central Romanian county of Brasov.
Eugen Tomac had been seeking to lead a government of technocrats, but lacked support from the parties in parliament.
"Eugen Tomac withdrew his mandate this morning and as such I nominate Adrian Vestea as prime minister," Dan said on Sunday.
"At the moment it is clear that a political (government) solution is the right one."
Parliamentary parties have previously said that a minority political government, without a permanent majority in parliament, would be better than a government of technocrats.
Dan is seeking to end a political crisis that has stalled policymaking, endangered access to EU funds and driven the leu currency to record lows.
Vestea will have 10 days to form a government and win a parliamentary vote of confidence.
Romania's next parliamentary election is not due until 2028.
It has never held an early election and analysts say the likelihood of one now is small as the opposition far right leads opinion surveys, significantly ahead of pro-European parties.
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