Jeremy Rockliff made the drive to Government House on Tuesday evening to meet with Governor Barbara Baker, five days after he lost the confidence of the parliament.
"Following their conversation, Her Excellency is now taking the time necessary to give due consideration to all available options," Government House official secretary David Hughes said in a statement.
"By the end of the week the premier will meet with Her Excellency again.
"Further statements will be made in due course."
Tasmania is staring down the barrel of its fourth election in seven years. The Liberals were returned to power in minority in March 2024 with just 14 of 35 lower-house seats.
Mr Rockliff has refused to resign after losing Thursday's vote of no-confidence.
"I respect the need for her excellency to take the appropriate time to consider important matters of state," he said in a statement.
"I remain committed to serving the people of Tasmania."
The no-confidence motion was put forward by the Labor opposition and supported by the Greens and three crossbench independent MPs.
Labor said the motion against Mr Rockliff was necessary because of his financial "mismanagement" and poor handling of delayed and over-budget new Bass Strait ferries.
The governor could ask the Liberals to find another leader who could fill the role of premier.
Labor, which has 10 seats, has ruled out forming a minority government in an alliance with the five-seat Greens plus members of the crossbench.
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