The Associated Press does not declare winners in races subject to an automatic recount.
Balderson's campaign manager did not dispute Kasich's claim Monday, instead declining to comment directly on it.
If Balderson's lead holds, his narrow victory in a seat President Donald Trump carried by an 11-point margin in 2016 would still a bad sign for the GOP, which will have to defend far more competitive seats than this one in the November midterms. He appeared equally uncomfortable answering questions from reporters about whether Trump's appearance might blunt the impact of the recent endorsement from a prominent Trump adversary, Republican Gov. John Kasich. "America is on the right path, and we are going keep it going that way". "Strong on Crime, Border & Military", the president tweeted on the eve of the election.
Voters are heading to the polls to decide key races including governor where seven Republicans and five Democrats are seeking their party's nomination.
In Kansas, Kris Kobach, a conservative leader in the movement to restrict illegal immigration and pass more restrictive voting laws, narrowly led Governor Jeff Colyer with about 84 percent of the votes counted for the Republican gubernatorial nomination.
Voters on Tuesday were choosing between Balderson, a state senator, and O'Connor, the Franklin County recorder, to complete the term of a Republican who retired in January.
Several voters casting ballots in suburban Westerville Tuesday, both Democrat and Republican, said they saw little difference between the two candidates. Republicans have controlled the district for 35 years.
The BBC correspondent in Washington says even if the Democrats lose the seat, they can claim a moral victory in OH, and will point to the results as further evidence that they are poised for success in November. Provisional votes remained to be counted.
It's unclear how much Trump's support helped or hurt Balderson.
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He instead chose to have his own ceremony earlier in the day at his alma mater, the University of Tennessee at Chattanooga. Owens has made the unprecedented move of deciding not to attend the inductions at the Canton shrine.
His opponent's campaign was described by some analysts as "not serious".
But US media said the race was still too close to call, in a seat long held by the Republicans.
According to data obtained by TIME, Balderson had only spent $507,206 on television ads as of August 3, while O'Connor had spent over $2 million.
In Michigan, three mainstream Democrats in suburban Detroit were among those vying for a chance at retiring Republican Rep. Dave Trott's seat in November.
Democrats, meanwhile, have been attacking Balderson by casting his support of tax cuts as threats to Social Security and Medicare.
"A Republican loss well might tell us more".
Republicans are scrambling to avoid an embarrassing defeat in a seat that the party has held for decades.
Senate contests in Missouri and MI were coming into focus as two vulnerable Democrats, Missouri's Claire McCaskill and Michigan's Debbie Stabenow, were expected to easily claim their party's nominations.
"Josh Hawley has established himself as a conservative champion for Missouri families, and we have no doubt he will beat Claire McCaskill this November", said Chris Hansen, executive director of the National Republican Senatorial Committee.