The Take That singer met Ferguson while holidaying in the Caribbean on John Caudwell's yacht.
He reportedly told her she is welcome at his home in Los Angeles at any time with her two daughters - Beatrice and Eugenie.
A source said to UK newspaper The Sun: 'Robbie and his wife Ayda really hit it off with Fergie. They were all on John Caudwell's yacht and chatted for hours about all sorts.
'Robbie's been telling everyone how much he likes her. He's told Fergie and the girls they can visit him and Ayda in LA whenever they like. Robbie says they'll be friends for life.'