Watling and captain Kane Williamson will have until Friday to prove their fitness.
Never before as any New Zealand team won as many Test series in a row as Kane Williamson's.
Bangladesh captain Mahmudullah was again left to rue a lack of application against pace.
If Williamson is ruled out, Will Young will make his Test debut at Hagley Oval.
Wagner claimed 5/45 and Trent Boult 4/52 to skittle Bangladesh on the fifth morning.
Taylor's 200 off 212 balls was his 18th Test hundred, taking him past Crowe's mark of 17.
At stumps on day four at Basin Reserve, Bangladesh were 80/3, still 141 behind New Zealand.
New Zealand declared with a 221-run first innings lead and 23 overs remaining in the day.
New Zealand plundered 334 runs off just 64 overs on Monday for the loss of one wicket.
Williamson injured his shoulder while fielding on Sunday and twice needed attention.