All of the complaints have been made within the past week.
Last week, the Orange Order has said the painting was a "deliberate demonisation of its cultural heritage".
On Monday, a delegation of DUP MLAs met staff from National Museums Northern Ireland (NMNI), including its acting chief executive Jude Helliker.
The DUP delegation included North Belfast assembly members William Humphrey and Nelson McCausland.
Christian Flautists Outside St Patrick's, by the late Belfast artist Joseph McWilliams, is being shown as part of the Royal Ulster Academy's annual exhibition.
After the meeting, NMNI said in a statement that it "welcomed the opportunity to listen to their concerns and engage in dialogue".
"We have taken steps to acknowledge that some have found the painting offensive and, last week, we placed a sign at the entrances to the gallery," it added.
The sign, situated at the entrance to the exhibition on the museum's fifth floor, reads: "Visitors may find some images in this exhibition thought-provoking, controversial and potentially offensive."
The DUP's William Humphrey said they had not reached "a meeting of minds" with the museum representatives during the meeting, but it was "important that we set out clearly our strong concerns".
He said the painting was based on an actual event, and he had been contacted by a constituent who was "appalled that he and members of his family are being depicted in this painting as being members of the Ku Klux Klan".
He said the museum had "acknowledged that there is a problem by erecting warning signs", but his party was "deeply disappointed that this derogatory painting will remain on display".
The Royal Ulster Academy exhibition has been running at the Ulster Museum since 16 October.
More than 35,000 people have visited the museum in that period, with about 4,000 visiting over the past weekend.
The painting has already been purchased by a private collector, but will hang in the museum until the exhibition concludes.
Six complaints have been made to the Ulster Museum about a painting that appears to depict a number of Orangemen wearing Ku Klux Klan clothing.