The last New Hampshire survey was conducted in mid-May by liberal outlet Public Policy Polling and showed Obama up by 12. Obama leads by 6 percentage points, according to the Real Clear Politics average of polls.
Romney launched his presidential bid and officially kicked off his general election campaign in New Hampshire. He also owns a vacation house and maintains close ties to the Granite State, which is near his home state of Massachusetts, where he served as governor.
New Hampshire is one of 12 battleground states that will be critical in determining the outcome of the 2012 election.
The Rasmussen poll of 500 likely voters was conducted on June 20 and has a 4.5 percentage point margin of error.