Neeson, who has also played gaelic football for Antrim, has signed a pre-contract agreement with the Reds.
The 27-year-old will move to the Solitude club in the summer when his contract with Carrick Rangers expires.
Striker Joe Gormley signed a pre-contract deal with the club in January after agreeing a three-year contract.
Gormley was unable to play for the club this season as he had already turned out for Peterborough and St Johnstone, on loan, during the present campaign.
Ex-Warrenpoint boss Gray was appointed as successor to Gerard Lyttle, who left to manage League of Ireland club Sligo Rovers last month.
Cliftonville finished fifth in the Premiership and lost out to Glenavon in the semi-finals of the Europa League play-offs.
Meanwhile Glentoran manager Gary Haveron has signed Corey McMullan from Ballyclare Comrades and John McGuigan from Warrenpoint Town.
Premiership new boys Warrenpoint have acquired the services of goalkeeper Alan Blayney, Darren Murray, Sean Mackle and Matthew Parker in preparation for their return to the top flight.
New Cliftonville manager Barry Gray has made Carrick Rangers goalkeeper Brian Neeson his first signing following his appointment to the role last week.