Chelsea signed Morata for a club-record fee of £60m ($78m) from Real Madrid in July as a replacement for Diego Costa.
The Blues turned their attention to the Spain international after Romelu Lukaku opted to join Manchester United in a £75m deal from Everton earlier that month.
Morata confirmed that United, who Chelsea face on Sunday, 5 November, and other clubs were interested in signing him, but added that manager Antonio Conte and members of the Blues board convinced him to move to Stamford Bridge.
"Antonio, [technical director] Michael [Emenalo], [director] Marina [Granovskaia], all of them called me this summer and there was no chance I wasn't coming to Chelsea," the 25-year-old was reported as saying by the Independent. "They demonstrated to me that they really wanted me.
"I had the chance to go to Manchester United and other teams in the Premier League.
"The important thing is I'm part of Chelsea and I'm very happy with this and now it's time to fight for the blue shirt. I want to play well and win with this shirt."
Morata has netted seven goals in 14 appearances for Chelsea this season, but the Spaniard has not scored for the Blues since the 2-1 win over Atletico Madrid in the Champions League in September.
The former Juventus striker threw his future at Chelsea into doubt last week after he said in an interview that he did not plan on staying in London for too long.
"What fascinates me about London is its multi-ethnicity, the coexistence of cultures and religions, but I do not see myself living here for very long," Morata was quoted as saying by the Daily Telegraph. "It's too big, too much stress, too much of a metropolis."
The 25-year-old later backtracked on his remarks, saying he would sign a 10-year contract at Chelsea if he was offered one.