A reported figure of around £9 million plus bonuses is said to have been received by the Robins after the Cherries saw two bids lose back last week.
Manager Nigel Pearson admitted in a press conference that Semenyo is now expected to travel down to the south coast for formalities ahead of putting pen to paper.
“At the moment, Antoine is in discussions with Bournemouth.
“I can confirm that. But the driver of the type of situation is that both clubs agreed on the fee. That has happened now.
“The club, at some point, will reveal what it is, but not today.
“He has been given permission to talk to the club and there will be an opportunity for a medical. That’s where we are with that.
“There is no more real comment about it now.
“The most important fact to recognize now is the fee that has been agreed between the two clubs, so now there are other factors to look at; one being medical and one agreeing to personal terms,” Pearson told reporters.
Semenyo is playing for the Black Stars at the 2022 World Cup in Qatar and has scored 20 goals and 22 assists for Bristol City.