The Legislature is considering a measure that would abolish capital punishment and replace it with a sentence of life without the possibility of parole.
Richardson said he would have vetoed such a bill just a few years ago.
"Right now, I'd say it's probably a 50-50 proposition," the second-term governor told The Associated Press.
"I'm struggling with my position, but I definitely have softened my view on the death penalty," he said.
A repeal bill passed the state House of Representatives last week and is pending in a Senate committee.
The governor said he was "sending a signal" to senators not to use him as an excuse to sidestep the issue.
"My main message is that it's critically important that the state Senate take a vote on it and not interpret my past position -- or my silence -- as a reason not to take up the vote," the Democrat said.
Richardson said he has been influenced by arguments of prosecutorial abuse and miscarriages of justice.
The alternative of life without parole "seems to me strong punishment," he said.
The cost of the death penalty, while low on his list of concerns, is "a valid reason in this era of austerity and tight budgets," he said.
Richardson also indicated he has been influenced by his discussions with Archbishop of Santa Fe Michael Sheehan, who presented the death penalty as a human rights issue. The Roman Catholic Church in New Mexico is an outspoken advocate of death penalty repeal.
Elected to a second term in 2006, Richardson is ineligible under the state constitution to run again in 2010. He abandoned his bid to be the Democratic presidential nominee last year after a poor showing in early primaries.
Obama appointed Richardson as his commerce secretary, but he withdrew after the disclosure that a grand jury is investigating allegations of wrongdoing in the awarding of contracts in his state.
New Mexico has two men on death row and has executed one person in the past 49 years, child killer Terry Clark in 2001.
"I have not made up my mind. I still may end up vetoing the bill," Richardson said.