Qatar's foreign ministry on Monday said that the case had been politicised and accused Manama of being "narrow-minded".
"The state of Qatar condemns the mentioning of its name in Bahrain's political disputes and internal conflicts," the ministry said in a statement.
It added that Bahrain should not resort "to the politicisation of the judiciary or use Qatar's name in favour of political calculation that has proven to be narrow-minded".
Shortly after, Bahrain's foreign ministry renewed warnings against "any form of interference or harm to the integrity and independence of the kingdom's judicial system".
Human rights groups have condemned the verdict against Salman, with Amnesty International calling it "a travesty of justice".
Bahrain and Qatar have been locked in a bitter regional dispute since June 2017, when a Saudi-led boycott of Doha was enforced over allegations it was cosying up to regional arch-rival Iran and supporting radical Islamist groups.
Ruled for more than two centuries by the Sunni Al-Khalifa dynasty, Bahrain has been hit by waves of unrest since 2011, when security forces crushed Shiite-led protests demanding a constitutional monarchy and an elected prime minister.
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