Davis has already added strikers Ryan Lowe and Chris Dagnall and Newcastle academy midfielder Liam Smith, but believes his squad is still lacking in numbers ahead of the new season.
"I still think we're short in certain areas," Davis told BBC Radio Stoke.
"A winger will be coming in but unless anyone leaves, it's unlikely anyone else will come in."
Although Davis was given an increased budget following relegation from League One last season, finances remain tight at Gresty Road when it comes to strengthening the squad further.
"I can't make the budget stretch that far," Davis, 50, said. "I can't bring anybody in because of the restrictions that I've got."
So for now, Davis will make do with the one new recruit who will add something different.
"He's quick, so he'll get us up the pitch and create problems," Davis said.
"All of our current wingers want to play deep and link the play, and it sometimes isolates the strikers, so we've got to find a way to give them better support with more crosses and shots."
From David Platt and Dean Ashton to Seth Johnson and Danny Murphy, the Alex have had a long-standing reputation of producing players that have gone on to play at a higher level.
Two more academy products - Grant Hanley and Ashley Westwood - could be the latest to net their former club a much-needed windfall due to sell-on clauses when they left Gresty Road.
Blackburn defender Hanley, 24, is a transfer target for Newcastle and Aston Villa midfielder Westwood, 26, has been linked with Premier League newcomers Burnley.
"If it was a possibility, that would be great," Davis said. "I know I can bring in someone who could do a really good job.
"The beauty of these sell-ons is the club gets rewarded and it gets out of a little hole."
League Two club Crewe Alexandra are set to sign a new winger, manager Steve Davis has said.