"We tried to go $24K for 24, but I told him, 'No, I'm going to keep my 24,'" Dixon told CSNBayArea.com on Tuesday.
Dixon has worn No. 24 since being taken by the 49ers in the sixth round of the 2010 draft.
"I worked hard to get that number when I first got here," Dixon told the website. "I worked hard to get it off somebody else. I've been No. 24 since high school and college."
Asomugha wore No. 24 the past two seasons with the Philadelphia Eagles but was released by the team this offseason. He asked Dixon if he could wear the number in training camp.
"When he first got here, he asked me what I thought of him wearing No. 24, and I said, 'Well, you can wear it,'" Dixon told the website. "I wanted him to look good. He asked if he could wear it so he doesn't look like a bum. Every other number was already taken."
Asomugha, who earned a roster spot as the 49ers' No. 3 cornerback, currently has No. 28.
Asomugha, meanwhile, had been limited in practice Wednesday with a collarbone injury. 49ers defensive coordinator Vic Fangio, however, said Thursday that Asomugha will be the 49ers' No. 3 cornerback for Sunday's game against the Packers.
ESPN.com 49ers reporter Bill Williamson contributed to this report.