Davis is understood to be looking for a new deal with the team as he enters the penultimate year of his six-year, $42.7m (£25.5m) contract.
However, his attempts to use his absence as leverage for a more lucrative deal will cost him $200,000 (£119,000) in workout bonuses for not showing up to the OTAs.
Davis is regarded as one of the 49ers' most important players, but the franchise is currently close to their salary cap limit, which could have an impact on the terms they are able to offer the 30-year-old.
Quarterback Colin Kaepernick and wide receiver Michael Crabtree could also be above Davis in the pecking order for new contracts, with both of their current deals running out at the end of the 2014 season.
Davis became the first tight end in NFL history to record two seasons with 12 or more touchdown catches last year, claiming 13 in addition to 850 yards.