Mukund has previously played for Chennai Super Kings and will be with the Bangalore team this year, RCB said.
Later in the day, RCB also announced the signing of Sunny Sohal, who was with Kings XI Punjab and Deccan Chargers.
"RCB have signed up batsman Abhinav Mukund as a part of the squad for the 2013 Indian Premier League," RCB said in a report on their website.
The 23-year-old left-hander has played five Tests for India and has led India 'A' teams. So far, he has played 15 Twenty20s scoring 235 runs.
This is the second signing by RCB in less than week. They had earlier picked 21-year-old Kerala paceman Sandeep Warrier, who had a good Ranji Trophy debut season 2012-13.