The president is on his way to Ohio to speak about economic prosperity, visit a Whirlpool plant, and attend a fundraising reception.
Mr DeWine has no symptoms and has returned to Columbus where he will quarantine for 14 days.
The Covid-19 test was administered as part of the standard protocol when meeting with the president.
Both the governor and first lady, Fran DeWine, who also has no symptoms, will be tested at home.
Ohio’s lieutenant governor John Husted also took a test today, which was negative.
The Republican governor has received bipartisan praise for his response to the coronavirus pandemic in his state — Ohio was one of the first to close bars and restaurants and implement a stay-at-home order.
Mr DeWine also advocated face masks, condemning the partisan debate over their use.
This is not the first time a governor has tested positive for Covid-19. Oklahoma governor Kevin Stitt announced that he had contracted the virus several weeks ago.
The president’s visit to Ohio comes as polling suggests that the state, which he won in 2016, is now in play in November’s general election.
Ohio has seen a surge in coronavirus cases since mid-June. There have been 96,305 confirmed cases of Covid-19 in the state, and 3,596 deaths.
On 5 August 1,199 new cases were recorded, down from a peak of 1,733 on 30 July.