The Electoral Commission (EC) said results had been delayed because supporters of the two main parties were impeding the process and it had asked the police to clear the collation centres.
Mahama’s supporters have taken to the streets to celebrate around the country.
Bawumia said he was basing his statement on the results according to internal tallies from the governing New Patriotic Party (NPP).
He said these showed Mahama had won “decisively”, while the opposition National Democratic Congress (NDC) had also won the parliamentary election.
President Nana Akufo-Addo is stepping down after reaching the official limit of two terms in office.
Mahama, 65, previously led Ghana from 2012 until 2017, when he was replaced by Akufo-Addo.
His previous time in office was marred by an ailing economy, frequent power-cuts and corruption scandals.
However, Ghanaians hope it will be different this time round.