The New York investment bank will double staff in the German city to 400 in order to be ready to continue to operate in Europe by the March 2019 deadline for Brexit to take effect, the person said.
"It's a short-term solution," the person said, adding that Morgan Stanley executives were considering other cities for its permanent EU headquarters.
Morgan Stanley currently employs about 5,000 people in London, but will relocate some traders as well as marketing and administrative staff to Frankfurt.
The bank also plans to shift a smaller number of banking and trader jobs to Paris, Dublin, Madrid and Milan, the person said.
French President Emmanuel Macron has been actively trying to lure bankers to Paris for the post-Brexit era, as have other European leaders.
Standard Chartered and Nomura already picked Frankfurt as their new EU headquarters. JPMorgan Chase has transferred hundreds of jobs to Dublin, but has not announced plans for a permanent headquarters for the EU.
Meanwhile, the US asset management company Northern Trust announced on Wednesday that it had chosen Luxembourg to create its "EU-banking presence."
The country's agency for the development of the financial center, Luxembourg For Finance, welcomed the announcement, saying Northern Trust joins "other leading financial institutions" in choosing the Grand Duchy "following the UK's decision to leave the European Union."