American Jewish organisations have called on the US President-elect to "address the wounds of an extraordinarily divisive contest".
The Trump campaign was widely criticised for using traditional antisemitic themes, such as attacking "global elites" for conspiring against ordinary Americans.
David Harris, head of the American Jewish Committee, urged Mr Trump to defend diversity "against any further attempts to demonise or stigmatise on the basis of ethnicity, race, gender or faith".
In Britain, a letter signed by over 100 people criticised Board of Deputies president Jonathan Arkush for being one of the first to congratulate Mr Trump soon after his triumph was confirmed.
Among other US reaction, David Bernstein, CEO of the Jewish Council for Public Affairs, which deals with relations with the wider community, called on the President-elect "to continue to assure the nation, particularly constituencies feeling most vulnerable, that the country will live up to its highest ideals and respect the rights of all people".
Matt Brooks, executive director of the Republican Jewish Coalition, offered a cautious welcome, saying: "Whether it's the Iran nuclear deal or the erosion of support for Israel in the Democrat Party, it's clear the American people are ready to turn the page on the past eight years". However, the coalition had been lukewarm in its support for Mr Trump and campaign donations had been slow.
One prominent board member - Ari Fleischer, White House press secretary under George W Bush - publicly disavowed Mr Trump close to polling day, claiming he had "lost control of himself and his message".
David Peyman, Mr Trump's head of Jewish outreach, told the Forward newspaper: "All I saw across America was fantastic support, support from pro-Israel voters [and] support from the American Jewish community."
He pledged that the President-elect would be Israel's "greatest friend" but refused to answer claims of antisemitism during his campaign, dismissing such concerns as "negativity".
The UK row followed congratulations offered by Board of Deputies President Jonathan Arkush.
He said: "I would like to congratulate Donald Trump on his victory. After a divisive campaign, I hope Mr Trump will move to build bridges and ensure that America's standing as a beacon of progress, tolerance and free-thinking remains strong."
Among a wave of negative responses on social media, Ivor Caplin, the former Labour MP and defence minister, said: "Arkush should have kept quiet but he seeks publicity instead of reflecting concerns of Jews."
Some Board members, including Amos Schonfeld, Liron Velleman, Ella Rose and Ben Lewis, as well as members of the Jewish Labour Movement and youth groups including Habonim-Dror, RSY-Netzer and Noam voiced similar sentiments.
Tal Ofer, who is on the Board's executive committee and defence division, tweeted: "I don't think it's @BoardofDeputies job to congratulate Donald Trump on his election, and I'm sure the Jewish community will agree with me".