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.
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 BBC 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".
"Hilary was a good secretary of state, things change, secretaries of state come and go - I know that – but you can have confidence in Chris coming in".
