Wilson is running as a slight underdog in the Democratic-leaning state. Her pollster, Glen Bolger, said she trailed Heinrich by 3 points in their latest internal poll, a result that was only slightly better for her than public polling of the race.
To win, she'll have to run more than a few points ahead of Mitt Romney in a state that has shifted strongly toward the Democrats in recent years, partly due to Hispanic population growth.
Wilson slammed Heinrich for not supporting the Keystone XL oil sands pipeline and criticized him for supporting the climate change legislation known as "cap and trade," arguing it would have been a major tax increase a weak economy. Heinrich has been aided by environmental groups that have already spent heavily on television ads against Wilson in the race.
She also criticized Heinrich for voting to "cut a half a trillion dollars" from Medicare by supporting Democrats' health insurance overhaul, and she called for the law's repeal.
But she avoided saying how she would have voted on this year's House Republican budget, sponsored by Budget Committee Chairman Paul Ryan (R-Wis.), that would have cut Medicare spending and largely privatized the program.
She pointed out that she'd voted against previous Republican budgets in the House, and while she commended Ryan for coming up with the overall plan, said she had "some concerns" about its Medicare provisions.
When asked what the concerns were, she said she hadn't looked at the bill since it was first introduced and didn't "have all the specifics" needed to state her position on it.
The Democratic Senatorial Campaign Committee hit back at her for her comments.
"If New Mexicans send Heather Wilson back to Washington, there is no doubt she would support cutting Medicare and privatizing Social Security in order to preserve tax breaks for companies that ship jobs overseas and give more tax breaks to millionaires," DSCC spokesman Matt Canter said.
The Hill rates the race "leans Democratic."
This post was updated at 2:17 p.m.