Senate Minority Leader Mitch McConnellAddison (Mitch) Mitchell McConnellWarren on family separation policy: Trump is ‘taking America to a dark and ugly place’ Senate GOP tries to defuse Trump border crisis Schumer rejects GOP proposal to address border crisis MORE (R-Ky.) maintains a significant lead over his primary challenger but is in a statistical dead heat with Democrat Alison Lundergan Grimes for November, a new poll suggests.

A Bluegrass Poll released Sunday by The Courier-Journal found 55 percent of likely Republican voters support McConnell. 

Just over a third support Matt Bevin, a businessman backed by the Tea Party, who will face McConnell in the GOP primary on Tuesday.

Three other Republican candidates split 5 percent of the vote, and 5 percent said they were undecided.

Looking toward November, the poll found 43 percent of registered voters in the state favor Grimes, while 42 percent back McConnell, who has served five terms in the Senate. 

About a third of voters approve of McConnell’s job in the Senate, compared to more than half who disapprove of it. 

Only 29 percent have a favorable view of McConnell, and nearly half have an unfavorable view. 

A majority of registered Republicans, the poll found, said they agree with Bevin’s argument that McConnell hasn’t done enough to stop ObamaCare or cut federal spending.

Despite McConnell's disapproval rating, half of voters say Bevin is too inexperienced. 

McConnell could have a tougher time in the general election. Thirty-nine percent of Bevin's supporters said they would back McConnell in November, but a quarter said they would cross party lines. 

McConnell’s Senate campaign also circulated a memo Monday in advance of Tuesday’s primary, laying out reasons why he is in a position to win.

“Senator Mitch McConnell is poised to win a historic primary election and enter the general election with significant momentum. In addition to fully implementing a more sophisticated campaign apparatus at an earlier date than ever before, McConnell has improved his standing amongst the general electorate significantly.”

The memo said McConnell has maintained a slight lead ahead of Grimes in 7 out of 11 polls released in 2014.

His campaign also downplayed a portion of the Bluegrass Poll that found a quarter of Bevin supporters might vote for Grimes in November. 

“In Kentucky primary politics, this is anything but unusual,” the memo said. 

The poll, a mix of automated calls and live-caller cellphone calls, surveyed 2,000 adults between May 14 and May 16, and has a margin of error of 4 percentage points.

— This post was updated at 9:20 a.m.