Mourinho parted company with Real a week ago and was presented as Chelsea coach on Monday but as the world and European champions prepared for a warm-up match against Ireland in New York on Tuesday, his presence was still being felt.
Spain defender Arbeloa, a supporter of Mourinho, has suggested the former Inter Milan boss was let down by members of the dressing room during his time at the Bernabeu.
Club and international team-mate Raul Albiol has spoken of a 'cooling' friendship between Arbeloa and goalkeeper Casillas.
The Spain number one, who was dropped from the Real side in the season run-in, was considered to have had a strained relationship with his coach and became a symbol for the 'anti-Mourinho' faction of fans at the Bernabeu.
"I haven't seen anything that has caught my attention," Del Bosque told a news conference when asked about the pair.
"They sit and eat at the same table, and if there is any disagreement, that is nothing strange because it happens in every team. There are 23 players and they all have their own opinions.
"Their relationship does not affect the overall health of the national team. If it isn't good at their clubs, that isn't my business."
Del Bosque could be forgiven for being pleased to see the back of Mourinho.
The escalation in the rivalry between Real Madrid and Barcelona reached new heights during Mourinho's three-year spell in the Spanish capital, which caused increased tension between the players when they met up for international duty.
More than half of Spain's squad comes from the world's two richest clubs by income, and Del Bosque was often forced to field questions about the harmony in the squad between the two sets of players.
The rivalry reached a peak after the teams played four bad-tempered 'clasicos' in 18 days at the end of the 2011 season, the games often shrouded in controversy, and then an explosive Spanish Super Cup clash at the beginning of the next season.
It is a new twist to the story that Del Bosque, himself a former Real player and coach, now has to field questions about disputes between players from the same team, even after Mourinho has gone.
On Sunday, another Spanish international, Barca's Andres Iniesta, gave an interview in which he said Mourinho, "damaged Spanish football".
By way of reply at his presentation on Monday, the new Chelsea manager responded: "If I have done some damage to Spanish football it is because I am the coach that ended Barcelona's hegemony."
The 50-year-old Mourinho left Real despite having three years left on his contract after finishing a season without a major trophy for the first time in his career.
Barca won their fourth La Liga title in five years.
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