Sinn Féin MLA Pat Sheehan said personalities were “irrelevant” regarding the appointment of the new secretary of state and that policy needed to change.
“The name plates may change around the British cabinet table,” he said.
“What’s important for us is that policy changes and particularly in regard to the full implementation of the Good Friday Agreement, specifically in terms of preparing for a referendum on Irish unity,” he said.
Social Democratic and Labour Party leader (SDLP) Claire Hanna MP, who had a meeting with Sir Chris on Tuesday, said there were two “immediate priorities” for the new secretary of state.
“The first is ensuring the executive agrees a budget before the 1 August deadline,” she said.
Hanna said the second priority was the “urgent need for institutional reform and calling time on the decay and dysfunctional politics that have held Northern Ireland back for far too long”.
TUV MLA Timothy Gaston told the Nolan Show that Sir Chris must be a “secretary of state that is willing to repair the damage done by the Irish sea border”.
“What we have seen in the past, over the last number of months is that he is unsympathetic to doing anything about that.”
Gaston cited Sir Chris’s previous role as minster for trade, when new steel tariffs were introduced earlier this month.
Shaun Woodward, a former Labour MP who served as Northern Ireland secretary in 2009-10, said he has known Sir Chris for more than 25 years.
“He is the kind of secretary of state Northern Ireland needs, I think it is a very astute appointment by the prime minister,” he said.
Speaking to Radio Foyle’s North West Today programme, Woodward said Bryant “is a very fair minded person”, adding he is a “fast learner, a very smart guy”.