The 28-year-old Western Australia offspinner is in a battle with Victoria's Todd Murphy to take over once Nathan Lyon retires from Test cricket and he made an early season statement in Darwin on the opening day of the three-day clash.
Rocchiccioli took 6-83 to help the Cricket Australia XI restrict Bangladesh to 263 at Marrara Cricket Ground.
He snared 38 wickets in 10 Sheffield Shield matches last season at an average of 28.18 for WA to continue the hot form he has displayed in the past three seasons.
Rocchiccioli's aim is to take the next step in his career.
"It is a dream. I want to play Test cricket and play for Australia but right now my journey and my stepping stone is ... playing what is in front of me," Rocchiccioli said after day one against Bangladesh.
"If you keep performing you get the right to have your name in the calculations.
"It was a different wicket to what I was expecting. I have played a lot of games up here and a lot of the time it is not as tacky or spinny early as it was today.
"I was quite happy when we got to bowl."
Former Australia captain Border told foxsports.com.au this week that he was concerned when Australia did not pick a spinner in the Test at the SCG last Ashes series when Nathan Lyon did not play.
"But I think that Rocchiccioli from WA ... he interests me because he gets people out," Border said.
"It's one thing to bowl well and get ones and twos (in the wicket column), but he's been getting fives and sixes bowling in Perth, so I think that is a good sign for him."
The Bangladesh innings was held together by a blistering and unbeaten 109 by star allrounder Mehidy Hasan Miraz, who faced just 141 deliveries.
The Australia XI side had a shocking start to their innings with openers Campbell Kellaway (0) and Sam Konstas (1) both failing.
The duo have a chance this summer to put pressure on the Australia top order but will be hoping for a chance to score runs in the second innings.
Kurtis Patterson (26no) and Josh Philippe (22no) carried the hosts through to 2-51 at stumps.
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