The Sheffield Wednesday striker spent most of last season on loan to Derby County.
Bullen remains in caretaker charge of Sheffield Wednesday.
Winnall has impressed since joining Derby County from Sheffield Wednesday in August.
Sheffield Wednesday travel to Brentford in the Championship on Saturday.
The Sheffield Wednesday caretaker manager sought advice from a former Sheffield United player who's now the coach of Everton's Under-23s.
Carvalhal's fate as Sheffield Wednesday manager could be decided after Saturday's visit of Middlesbrough.
One Twitter account has linked Sheffield Wednesday with a move for the Walsall attacking midfielder.
The Sheffield Wednesday striker sparked speculation he could be en route to Hibernian this summer.
Zungu has been strongly linked with a move to Sheffield Wednesday over the past seven days.
Sheffield Wednesday are being linked with a move for the 24-year-old, along with Portuguese giants Benfica and Porto.