Full-back Vrsaljko, 24, has spent the past two seasons at Sassuolo in the Serie A and will make the move to Spain.
The 19-time international was also linked to Napoli, but opted for La Liga and European giants Atletico.
His agent, Giuseppe Riso, said he expected the transfer to be confirmed shortly.
"Soon, the move will be official," he told Radio CRC.
"Sime never rejected Napoli. I want to clarify that one thing.
"Napoli have made an important offer in January and now, but it is difficult to turn down Atletico, twice a finalist in the Champions League.
"It's not about the money or anything. He has been thinking a lot, I assure you, and only recent results have affected him."
Vrsaljko made 35 appearances in the Serie A last season, while he featured twice for Croatia as they made the last 16 at Euro 2016.