The Conservative People's Party, which has just six seats in parliament but whose support is crucial to the minority centre-right government, said it had lost confidence in Environment and Food Minister Eva Kjer Hansen.
“I have said (to Rasmussen) that I support the government and wish it to stay but Løkke has a minister that we no longer trust,” Conservative leader Soren Pape Poulsen told public broadcaster DR in remarks widely seen as a threat to the eight-month-old government.
At a press conference later, Rasmussen said Kjer Hansen still had his confidence and that sacking her was “not an attractive option.”
“It is a question of trust among the parties right of centre. But we are in the middle of a refugee crisis, and other challenges of maintaining our welfare system,” said Rasmussen.
Calling new elections less than a year into the government's mandate was also a “bad” option, he said.
“We also risk losing power. Not attractive,” he said.
Describing the crisis as Denmark's “hour of destiny”, Rasmussen said he was “reaching out” to Conservative leaders to try end the standoff.
Later on Wednesday, he was to chair a meeting of his ruling centre-right Venstre party and its allies to discuss the controversy around Kjer Hansen.
The Conservatives have accused the minister of giving them wrong information about a proposed package of agricultural regulations, which they say could have serious consequences for the environment.
Kjer Hansen's critics specifically accuse her of giving into the farm lobby on norms governing the use of fertilisers, leaving water supplies exposed to increased pollution from agricultural runoff.
'Battle of nerves'
Poulsen's threat to withdraw support for the government is nonetheless seen as a risky gamble, with analysts saying voters could punish the Conservatives for his political hold-up if it forces them to return to the polls.
It's “an explosive crisis and nobody knows what is going to happen… It's a battle of nerves,” leading political analyst Thomas Larsen told AFP.
Apart from the Conservatives, Rasmussen's Venstre party, which took office in June 2015, has the backing of the anti-immigrant Danish People's Party and the pro-business Liberal Alliance. Together, Venstre and its allies have a total of 90 seats in the 179-member parliament.
Larsen outlined three possible ways in which the crisis could unfold.
Either Rasmussen resists the calls to fire Hansen, which could bring down the government and trigger new elections, or Poulsen backs down, or Hansen resigns, which would solve the problem for the government, Larsen said.
In a sign a compromise could be in the offing Poulsen indicated he would not oppose Kjer Hansen being named to another ministerial portfolio if Rasmussen reshuffled his team.