But Tom Lindgren of Bonhams & Butterfields says the auction house is talking with a number of institutions and individuals, and is confident a sale will be completed in the next couple of weeks.
The auctioneer had hoped Saturday's bids would top $6 million for the T. rex dubbed "Samson" at the auction.
Lindgren says the highest bid today was $3.7 million.
Experts say the 170 bones discovered about 17 years ago in South Dakota represent more than half the skeleton of a 40-foot-long, 7.5 ton dinosaur that lived 66 million years ago.
A similar t. rex fossil sold for $8.3 million in 1997.