The deal is believed to be worth more than £400m which will also mean that they will become the 10th club in the Premier League under majority American control.
According to reports, the group had initially reached an agreement with Moshiri on 23 September and was awaiting regulatory approval, with the Premier League having the final say on the deal, which has now been granted.
The Friedkin Group is currently owned by chairman Dan Friedkin who also owns Serie A club Roma. The last report from Forbes confirms that Friedkin has a net worth of £6.16bn.