This was possibly in anticipation of his less-experienced colleague James Shaw entering the race.
Mr Shaw, who entered Parliament in September, said it was "highly unlikely" he would run for the job though he would consider it if he was given a strong signal by party members.
The Wellington-based MP cited inexperience and an eagerness to make progress with his new economic portfolio as reasons for not running.
Mr Shaw had already received a few calls urging him to compete for the role, though he admitted it would be "very hard" to beat Mr Hague.
He said he would make a final decision when nominations for the job opened on March 20.
Fifth-ranked MP Gareth Hughes, who became an MP in 2010, could also be a contender for Dr Norman's job.
Mr Hague is ranked third on the party list and holds the health, housing and ACC portfolios. He played a major role in the passage of same-sex marriage legislation in 2013.
Before entering Parliament in 2008 he was chief executive of the West Coast District Health Board and an executive director of the New Zealand AIDS Foundation.
Nominations for the position close in mid-April. After a series of public meetings around the country, delegates will vote on a new leader at the party AGM in May.