The north Wales club have released nine players following a disappointing season which saw them finish 13th in the National League.
"Recruitment is going to be key and massive throughout the summer," Keates told BBC Radio Wales.
"With the ones that we've already got we're probably looking at adding at least another 13 or 14."
Keates added: "It's getting the right people in the building with the right work ethic.
"I've been making phone calls over the last number of weeks and it's going to be even more of a busy period now.
"It's all hands to the pump to get the players in the building."
Midfielder Paul Rutherford has agreed a deal he was offered in February with Mark Carrington having already signed a new contract.
Keates has offered new deals to goalkeeper Chris Dunn and midfielder Russell Penn and said he was "fairly confident" they would sign.
He also wants to extend the stay of defender James Jennings, who joined the club on loan from Cheltenham Town in January.
"James is still contracted to Cheltenham," Keates said.
"That's something I've chatted with James about and obviously what develops with that over the next few weeks regarding what happens with him at Cheltenham."
Curtis Tilt, Hamza Bencherif and John Rooney will not be returning to the north Wales club having finished the season on loan at other clubs.
Iffy Allen, Khaellem Bailey-Nichols, Anthony Barry, Rob Evans, Shwan Jalal, Izale McLeod, Callum Powell, Martin Riley, Jordan White have been released.
"It needed to be done as soon as possible," said Keates, in his first managerial role after succeeding the sacked Gary Mills in October 2016.
"I came to the decision it was best to do it face to face and be open and honest as I could with the players.
"For me to take the club forward, difficult decisions needed to be made."
Club captain Evans was among the players released and Keates admitted it had been a difficult decision to let go of the 21-year-old Wrexham-born midfielder
"That was by far the hardest," Keates said,
"I've known Robbie from playing here and watched him come through the youth team and establishing himself in the first team.
"It was a difficult decision regarding Robbie."
Manager Dean Keates hopes to sign up to 14 players this summer as he rebuilds Wrexham's squad.