Lafferty has signed a two-year deal with the Ibrox club for an undisclosed fee and Budge revealed he had his mind set on the move.
Budge stated she reluctantly sold the forward who scored 19 times for Hearts last season. The deal should allow manager Craig Levein to strengthen Hearts' squad further before the transfer window closes.
"It is no secret that this has been a particularly difficult and drawn-out affair," Budge told the Hearts website. "While we have no desire to sell our players to other teams in the league, when a player has his mind fully set on moving, it is difficult to reach an alternative conclusion.
"The deal, when completed, will suit all parties and, importantly, will allow Craig to focus his efforts on finalising potential deals to add to his playing squad - having brought in 13 players already this summer - and preparing the team for their Ladbrokes Premiership match at Kilmarnock, where they will bid to go eight games unbeaten.
"It will also allow me to return my attentions to progressing the development of our new, multi-million pound main stand and improving the matchday experience for our supporters."