The 23-year-old Argentina international has made just eight appearances for Atletico since joining the Madrid outfit from River Plate in December 2015.
"We are negotiating (with Atletico) and we are very close to signing Kranevitter on loan," Sevilla sporting director Monchi told Spanish radio station Cadena Cope. "We expect to have developments this week."
Monchi is looking to replace Polish midfielder Grzegorz Krychowiak, who joined coach Unai Emery in making the move from Sevilla to Paris Saint-Germain this month.
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Krychowiak joined Sevilla in 2014 on a 4.5-million-euro (3.83 million pounds) deal from Stade Reims.
"It was a very important offer that we received for Krychowiak," Monchi said. "It was a very good price that we got for him, seven times what we paid for him two years ago. We will reinvest that money."
Sevilla, under new coach Jorge Sampaoli, take on Champions League winners Real Madrid in the UEFA Super Cup on Aug. 9 in Trondheim, Norway.
(Reporting by Adriana Garcia; Editing by Clare Fallon)
