The 56-year-old Murphy has been with the Padres' organization since 2010 and has managed the Padres' triple-A affiliate in El Paso the last three seasons.
"It's definitely an opportunity that he's earned," said Padres General Manager A.J. Preller. "I think it's a chance for him to go out, the team to go out, and show what he's capable of."
Murphy's record with El Paso over the last three seasons is 180-172.
Rodriguez tied Willie Mays for fourth place on the career home run list on May 1, and baseball's labor contract sets a 45-day deadline to file a grievance. MLB and the union said the deadline is on hold for as long as the two sides agree.
At the time Rodriguez and the Yankees signed their $275-million, 10-year contract in December 2007, they signed a separate marketing agreement that called for $6 million each for up to five milestone accomplishments.
The accomplishments were contemplated to be Rodriguez hitting home runs 660, 714, 755, 762 and 763. In exchange for each designation, Rodriguez would give the Yankees marketing rights, such as using Rodriguez's name and image in selling licensed goods.
Following Rodriguez's return this year from a season-long drug suspension, the Yankees said the decision to designate any historic achievements was at the team's discretion and they would not designate any.