Real were unable to end the dominance of great rivals Barcelona in Mourinho's first campaign in charge last season and have already added midfielders Nuri Sahin and Hamit Altintop, defenders Raphael Varane and Fabio Coentrao and forward Jose Callejon to their roster.
Reports in local media have linked the big-spending club with forwards Sergio Aguero of Argentina and Neymar of Brazil, currently representing their countries at the Copa America.
"We decided to improve and balance the squad without ever losing the dimension of creating a team for the future," Mourinho said in an interview on Real's website.
"We have signed players whom we felt were important to complete a squad that was already very strong," the Portuguese added.
"In my opinion, and in that of everyone, we are missing one more man, one more attacker."
Real's players and technical staff arrived in Los Angeles on Tuesday on the first leg of a pre-season tour that will also take them to Germany, England and China.
They play David Beckham's Los Angeles Galaxy on July 17 and Mourinho is due to appear at a news conference with the former Real player and England international later on Tuesday.
Real forward Cristiano Ronaldo, captain Iker Casillas, Galaxy coach Bruce Arena and captain Landon Donovan would also attend, Real said.
Mourinho said the low average age of the current Real squad meant it was good for the next 10 years.
"We have a team that needs to grow and at certain moments it could feel some weakness on an emotional level and in terms of experience," he said.
"I am delighted to work with them because they are highly motivated and want to learn and win. The technical staff is absolutely delighted."
