When reached for comment by MLB.com on Sunday night, Atkins respectfully declined to discuss the situation.
LaCava was hired by Shapiro in 2002 as a national crosschecker for Cleveland, but he left later that year to take the job as assistant GM for the Blue Jays. He has spent 13 years in the Toronto organization.
Atkins, meanwhile, has spent 15 seasons with the Indians and was promoted to VP of player development by Shapiro last offseason. Earlier this year, he also interviewed for the Angels' GM job, which went to Billy Eppler.
Atkins declined comment out of respect to both teams.