Kinnings said that players must now make a choice.
"We have rules and regulations which are there to protect our tour and which our members must all abide by. Players who are not willing to comply with our rules and regulations should not be seeking to join our tour," he added.
LIV Golf was not immediately available for comment.
Some top players were given conditional releases by the European Tour in the past, which allowed them to compete in both.
Membership of the European Tour is a requirement to play for Europe in the Ryder Cup.
Former world number one Jon Rahm has said he will not be participating in LIV Golf, which is fighting to emerge from Chapter 11 bankruptcy after Saudi Arabia's Public Investment Fund withdrew financial support and has secured funding from BC Partners Credit.
LIV has agreed to terminate Sergio Garcia’s contract, and Bryson DeChambeau, Cameron Smith, An Byeong-Hun, Marc Leishman, Cameron Tringale and Matthew Wolff are seeking release from old deals.
The BBC said it had been told DP World Tour will not accept membership from anyone intending to play in the latest version of LIV.
It added that the tour would have grounds to terminate the membership of anyone who did not disclose their intention to play with LIV and then played.
