Essential Edinburgh is now preparing an application for an order to allow court officers to be brought in to close down the camp.
The city centre business group cited public health concerns among other issues after finding human excrement within the gardens.
Official letters were delivered to each tent warning members will be evicted if they fail to leave the site.
A court action is expected to begin midweek at the Court of Session and the eviction carried out the following week if a judge grants the order.
Police would be deployed at the site to ensure public order while court officers physically remove camp members if they refuse to leave.
Essential Edinburgh will have to foot the bill for legal fees and pay the cost of bringing in the officers.
Some estimates have suggested the bill could be in the region of £10,000, which the business group could attempt to pursue the protesters for.
Legal letters served to the occupants said some members had “acted in an aggressive and antisocial manner” and cited incidents of vandalism.
Occupy Edinburgh – which has been camped on the square for three months – expressed dismay at the move, highlighting the fact that the site has been cleaned up.
It also insisted those who had exhibited antisocial behaviour had been expelled.
Andy Neal, chief executive of Essential Edinburgh, said the legal action was implemented after lengthy talks with the camp.
He said: “I’ve met with them to hear what their proposals are and suffice to say it still revolves around having a very significant presence of tents, which is the fundamental cause of our problem.
“The space was not designed to have people living in it and however hard they try to move tents around and discipline people for urinating against trees, it’s not going to work out.
“What we have seen recently, sadly, is an escalation in antisocial issues, which mean we now feel obliged to bring this to an end.
“We still hope they will recognise our request and voluntarily leave.”
Conservative city centre councillor Joanna Mowat added: “There is now a public health issue and a public protection issue, and quite simply the square has to be cleared. It’s a shame because this could still be done in a less confrontational way, but the commitment to give the square back to the people of Edinburgh has to be backed up with action.”
Members of the camp have so far pledged to remain at the site until they are forced to leave. However, it is understood that some believe that it could transfer to another site.
Chris Sharp, 24, a copywriter and camp member, from Edinburgh, added: “We are disappointed Essential Edinburgh isn’t willing to negotiate and recognise the moves we’ve taken to improve the square.
“We have thrown people out of the camp and those who have violated the rules.”