Bullock initially agreed to a two-year, $21 million deal, but that contract is being reworked as a result of an unknown medical issue. Morris had previously agreed to a two-year $20 million deal with the San Antonio Spurs but is considering backing out to take a deal with New York.
The revised Bullock contract would likely clear enough salary cap room to fit a player they have "long been enamored" with in Morris.
Chris Haynes of Yahoo Sports reported the Knicks have offered the forward a one-year, $15 million deal.
Although Morris hasn't signed a contract with the Spurs, the team completed other moves to fit him under the cap, including trading Davis Bertans, per Bobby Marks of ESPN.
Backing out of the agreement leaves the Spurs empty-handed without many other comparable options left on the free-agent market.
"[Spurs coach] Gregg Popovich has to be blowing a gasket," a source told Marc Berman of the New York Post.
It could also lead to mistrust in future negotiations involving Morris and potentially other players around the league.
However, it appears this will end favorably for the Knicks, who would be adding a starting-caliber power forward to a deep rotation down low that already includes young prospects Mitchell Robinson and Kevin Knox as well as recently added veterans like Julius Randle, Taj Gibson and Bobby Portis.
Morris averaged 13.9 points and 6.1 rebounds per game last season, knocking 37.5 percent from three-point range.
Meanwhile, Bullock's medical prognosis will help determine whether the team signs him at a fair price. If healthy, the guard can spread the floor as a 39.2 percent career shooter from three-point range.