"To conduct the negotiations" means to oversee or manage the process of discussing and reaching an agreement or settlement between two or more parties involved in a particular matter or dispute. Full definition
Must be skillful in presentation with strong ability in conducting negotiations in a persuasive manner. (jobdescriptionandresumeexamples.com)
Instead of conducting negotiations by letter or telephone, all parties will work things out in face to face meetings, with your solicitor providing support and guidance throughout. (jacksonlees.co.uk)