Under the deals, which are likely to be announced before the post-season, Fox will retain rights to the World Series and to a league championship series every year, the person said, speaking on condition of anonymity Wednesday because no announcement has been made.
The amount baseball receives from the two networks is likely to double to an average of about $800 million annually, with Fox's share averaging about $500 million.
ESPN and MLB last month announced a new deal covering 2014-21 that will increase ESPN's average yearly payment from about $360 million to approximately $700 million.
Fox, which broadcasts a Saturday regular-season game each week under its current deal, will gain additional regular-season rights under the new contract. Fox is expected to put some games on a national cable network that likely will be a rebranded form of its Speed network.
Turner also will broadcast an LCS each year, two division series and 13 regular-season Sunday telecasts. However, it will gain more co-exists with club telecasts and increased digital rights.
Turner had carried all four division series from 2007 through last year but gave up two division series games to the MLB Network under a deal running through 2013.
That was part of a financial agreement that gave it rights to the two wild-card round games this year. ESPN gains a wild-card game starting in 2014. It also had televised 26 Sunday games each season.
Fox broadcast the World Series in 1996 and 1998, then took over exclusive rights in 2000.