The England midfielder joined the club in January, signing from Al-Ettifaq on a free transfer.
After making his debut against PSV last weekend, Henderson is reportedly now set to be handed a huge honour by the Dutch giants.
According to the Mirror, he will be named the club's new captain just three weeks after signing.
Henderson previously spent eight years as Liverpool skipper from 2015 until his exit from Anfield last summer.
Fans were not impressed by Ajax's reported plans, though, with one writing on X, formerly Twitter as cited by Sports Central: "Embarrassing for such a respected club like Ajax."
Another added: "So disappointing."
A third supporter also commented: "Standards have really dropped at Ajax."
However, others were supportive of the mood as one wrote: "Because he is a real leader."
Meanwhile, another fan said: "He deserves it. Proven leader."
Despite the reports, Henderson was not handed the armband against Heerenveen on Sunday.
Although the 33-year-old started the 3-2 away defeat, Steven Berghuis was named as captain for John van't Schip's side.
Club captain Steven Bergwijn is currently sidelined with a hamstring injury after being given the armband in the summer.
Sunday's loss was Henderson's second appearance for Ajax after his January arrival.
Following a six-month spell in Saudi Arabia with Al-Ettifaq, he returned to Europe on a free transfer to join the Eredivisie outfit.
Ajax are currently fifth in the table, 21 points adrift of league leaders PSV and nine off the Champions League places.