The Blues have agreed deals with Argentina based River Plate for the defender, and their Uruguayan namesakes Club Atletico River Plate for the forward, with the defender set to join for around £6million and the defender expected to cost around £500,000.
Both players were guests at Goodison Park on Sunday for the game against Manchester City and have since been at Finch Farm.
Mori in particular will be a deal that could go right to the wire as the club look to secure international clearance for him, but they envision no problems on that front.
And manager Roberto Martinez was full of optimism in his Friday morning press conference.
"We're really, really excited. Ramiro was in the building yesterday. We hope to complete the deal sometimes today."
"International clearance could take a few days. Our relationship now with River Plate is very good and we're confident Ramiro could become an Everton player very soon.
"Leandro will be announced today. He is a young man who can make a goal out of nothing. He has an impressive record."
The double deal will come as a big boost for the Everton faithful, who have so far been left worrying that the club have only signed two players in this transfer market, with less than a week left of the window.
However, it seems the Blues will once again be doing their business late, with the deal for the two South Americans potentially one of a few late flourishes in the market.