There will be no announcement before the Senate returns on 22 February, said deputy press secretary Eric Schultz.
Republicans have demanded President Barack Obama - a Democrat - leave the nomination to his successor next year.
Mr Scalia, 79, was found dead at a Texas ranch on Saturday. He died of natural causes, a judge has said.
Presidio County Judge Cinderela Guevara told the Associated Press news agency she came to that conclusion after speaking to Mr Scalia's doctor and the police.
Why is this a big deal?
Scalia - in his own words
The death of a powerful conservative voice on the bench of the country's highest court threatens to spark a constitutional crisis in the US.
According to the constitution, the president nominates justices to the court and the Senate - currently controlled by the Republicans - uses its "advice and consent" powers to confirm or reject that person.
In recent years, the court has made key rulings on gay marriage, abortion and Mr Obama's key healthcare legislation.
The loss of Mr Scalia means the make-up of the court is now four justices picked by a Democratic president and four by a Republican, so there are four largely liberal voices and four largely conservative.
The conservatives have been in the majority but a new judge could tip the balance, and Republicans running for president are strongly opposed to Mr Obama making his choice.
Senate Majority Leader Mitch McConnell agrees - he believes such a key appointment should not be made by a departing president.
But the White House has vowed to press ahead, although there will be no swift decision within days.
"Given that the Senate is currently in recess, we don't expect the president to rush this through this week, but instead will do so in due time once the Senate returns from their recess," said Mr Schultz.
The presidential election is in November, with a new US president to be inaugurated in January 2017.
A new US Supreme Court judge to replace justice Antonin Scalia will be nominated by the president after next week, the White House has said.