The two sides are believed to have agreed a fee in the region of £11m for the 23-year-old, the the rest of the details of the move expected to be formalities.
Should the deal go through, Murphy will become Cardiff's most expensive signing since they were relegated from the English top flight back in 2014.
Murphy came through the academy at Norwich along with brother Jacob, who completed a similar move to Newcastle last summer.
In total, Murphy has made 108 appearances for the Norfolk-based side, enjoying a strong 2017/18 campaign that saw him score 11 goals in 47 games in all competitions.