Grant wants to bring the Dutchman to Stamford Bridge as one of two assistant managers and a deal is expected to be completed in the next two weeks.
Yesterday the two clubs and Ten Cate were all positioning themselves for negotiations.
Ten Cate, who is attracted by a move which would double his £1million salary, claimed he was not about to quit Ajax.
Grant explained why he wants the 52-year-old coach, who has 20 years of coaching and managing experience, culminating in his time as assistant manager at Barcelona and the past year in Amsterdam.
"It is very simple," said the Chelsea boss. "We are looking for the best. It's no secret we have lost four from the staff - one assistant, one fitness coach, one scout and one goalkeeping coach - and we have checked all the possibilities. It is easier to do it before the season but we are handling the situation very well and we will find them. I want to do it as soon as possible.
"Henk has a very good record as assistant at Barcelona. I have met him, he is a nice person and I like his style of football. First I decide how I want the team to play and then we look for the right person."