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Heffernan, who is currently back in Ireland spending time with family in the pre season period, is expected to travel to Ashton Gate as early as Monday to discuss personal terms and undergo the formality of a medical.
The six-figure sum agreed with the Coca Cola League One side has been confirmed as an initial £125,000 on top of a £25,000 payment depending on performances for his new club.
BCFC chairman, Lansdown told their official site: "We are very confident of clinching the deal now that we have agreed a fee with Notts County."
"The player will fly across to England on Monday and I'm sure we will be able to agree personal terms with Paul."