The 31-year-old Broad went past Botham's haul of 383 wickets during England's first-test win over West Indies by an innings and 209 runs at Edgbaston on Saturday.
Broad, who has 107 caps, bowled Shane Dowrich to claim his 384th dismissal and finish with five for 81 in the match.
"I have a lot of cricket left in me," Broad told reporters. "I'm really enjoying being a part of this team and hopefully have a few more miles in the tank.
"I'll be 33 in 2019. I certainly hope that my performances will keep improving so that I'm part of that 2019 Ashes."
England, who hold the Ashes, visit Australia for a five-test series in November. The two sides meet again in England in 2019.
Broad is keen on maintaining his fitness levels ahead of their trip down under but insists he would not want to be rested for the remaining two tests against the Caribbean side.
"This winter is a hugely exciting one," he added. "The two teams are very similar so it should be a belter.
"I will politely be suggesting that I will be having sufficient rest in September and October to be available for the next two tests against West Indies."
England's second test against West Indies begins on Friday at Headingley.
First Published: August 22, 2017, 9:50 AM IST