The 24-year-old striker was heavily linked with a move to Elland Road in the recent transfer window, but ultimately joined Aston Villa on a deal worth up to £33m.
Watkins has made an impressive start for the Villans, scoring five goals and providing one assist from seven appearances across all competitions.
Former Leeds star Beckford is dejected that the Englishman did not sign for the Whites and explained he spoke to Watkins over the summer.
He insists a deal to take Watkins to Leeds was close to happening and believes the striker would have fitted in well under Marcelo Bielsa.
“Their [Villa’s] summer window has actually been really good“, Beckford said on The Phil Hay Show.
“They have also got Ollie Watkins, another player that we were linked heavily with as well, somebody that I’d spoken to over the summer.
“He was on the cusp of joining the club, but for one reason or another, his move broke down and fell through.
“It was a shame if I am honest with you because he is not only a really good football player and a great goalscorer, but he is a lovely person as well.
“He would have worked out well for us.“
Three of Watkins’ five goals this season have come in the Premier League, with all of them being scored against Liverpool in Aston Villa’s 7-2 win.
Leeds will be keen to make sure they keep Watkins quiet this evening when they play Aston Villa.