Maradona was annoyed with Bilardo during the 2010 World Cup qualifiers over the alleged interference of the technical director in team affairs.
"For me it's not easy to be in the middle of an internal (conflict) between Diego and Bilardo because they are two people I care for," the 34-year-old Veron told reporters at the presentation of the shirt Argentina will wear at the World Cup finals in South Africa.
"But it's not a problem because I've always managed to keep on the fringe ... they are two men who resolve their issues in their own way, as they see fit, and I don't get involved."
Maradona handed Veron a key midfield role in the qualifiers which Argentina struggled to get through, needing to beat Peru and Uruguay in their final matches to avoid missing the finals for the first time since 1970.
The coach did not appear entirely happy, however, after Argentina's 1-0 win in Uruguay when Veron spoke of mistakes and the need not to "sweep them under the carpet" in the euphoria of qualification.
Veron repeated his concerns on Thursday, but also appeared to back the inexperienced Maradona.
"I think and hope there will be modifications because I know the self-criticism is there. Obviously we hope changes are made...for the good of everyone.
"There are cases of coaches with no experience in which things went very well for us (at Estudiantes), as happened with (Diego) Simeone or (Alejandro) Sabella.
"You just have to give them time."
Former Argentina midfielder Simeone coached Veron's Estudiantes to the league title in 2006. Former national assistant coach Sabella took charge of the team this year and steered them to their fourth South American crown in July.
Bilardo played alongside Veron's father Juan Ramon in the Estudiantes side that won three successive Libertadores Cups from 1968 to 1970.
