Lemar is now away on global duty with France and both clubs will be keen to finalise the transfer before the start of the World Cup, which gets underway on Thursday.
Griezmann, 27, continues to be linked with a move to LaLiga champions Barcelona after another fine season at Atletico.
"Over the next days, both clubs will work to close the agreement for the definitive transfer", a statement on Atletico's website read.
Despite Morata's best efforts, he and the rest of the world will have to wait a little bit longer to discover Griezmann's decision over his future.
Having previously stated that he would announce his decision before France's opening World Cup game on Saturday, Griezmann chose to keep quiet.
SGSpanish journalist Pedro Morata came up with a sneaky to ask Griezmann about his future
He's now preparing for the World Cup with France but given how far along the Liverpool transfer was, there's every chance his head is still on Merseyside.
He has scored three goals in 12 matches with the national team since his November 2016 debut.
Marca reported Atletico would pay between €70-80 million for the 22-year-old, and that he would sign for five years.