A Magellan Strategies poll released Thursday found Grimes leading McConnell 49 percent to 46 percent in the 2014 race. Five percent of respondents were undecided.
The poll also found that 58 percent of respondents would be more likely to oppose a candidate for U.S. Senate that supports the Obama administration's new carbon emission regulations, which seek to reduce emissions by 30 percent from existing power plants by 2030.
Grimes has distanced herself from the President on coal and released a statement promising to "fiercely oppose" those new carbon emission regulations. She has also made a point of criticizing McConnell for not doing enough to protect Kentucky coal jobs.
The poll was conducted June 4-5 using automated interviews among likely general election voters in Kentucky. Forty percent of respondents identified as registered Republican voters, while 56 percent identified as registered Democrats. It had a margin of error of 3.45 percentage points.