Pietersen was the third-highest run-scorer in the recent Big Bash competition in Australia, scoring more half-centuries (four) than anyone else.
"Can a handshake take place, a mature conversation?" said Vaughan.
"Or has he burnt so many bridges a new set of management couldn't handle him over a three-week period? He could be the X-Factor."
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England will take part in the World Twenty20 in India between 8 March and 3 April.
Pietersen, speaking before last summer's Ashes series, said he was "angry and hurt" after director of cricket Andrew Strauss told him he was excluded because "trust had eroded".
The 35-year-old batsman last played for England in January 2014 and later released an autobiography criticising some of his team-mates, and officials at the England and Wales Cricket Board.
However, he has vast experience of playing in India from his time in the Indian Premier League and starred as England won the World Twenty20 in 2010.
In the Big Bash, he scored 323 runs for Melbourne Stars including 62 in the semi-final and 74 in Sunday's final.
Asked if the selectors would pick Pietersen, former England captain Vaughan added: "They won't but I would because of the way that he's playing and the level of performance that he's showed.
"In front of massive crowds in Australia his batting won the semi-final and almost the final.
"A year ago he wasn't playing well enough and he was nowhere near the England team. But the T20 World Cup starts in a few weeks time and I personally want the best team out there for England."
England's opening World Twenty20 match is against West Indies in Mumbai on 16 March (09:30 GMT).

The memorial event at Whitecrook Park in the town was arranged on social media by friends of the 15-year-old who was found dead on Monday.
She was last seen alive near a deli in Clydebank on Saturday 19 March after she had set off for her part-time job at a hairdressers.
A man has appeared in court charged with her murder.
About 300 people braved wind and rain to assemble at the park, many of them carrying pink balloons in tribute to Paige.
Some of the teenager's friends addressed the crowd to thank them for attending in memory of a "most outstanding, amazing girl".
One girl said: "Paige was so beautiful, everything about her you just wouldn't change. For that to get taken away from us was absolutely heart-breaking."
Another added: "I just can't believe she has gone, it hasn't sunk in. There's only one star in the sky tonight and it's obviously Paige looking down on us."
A minute's silence was held before people lit candles and tied pink balloons to railings.
Some of those gathered played songs and held up posters dedicated to Paige.
On Thursday, the landmark Titan Crane in Clydebank was illuminated in pink in memory of the teenager.
At a private hearing at Dumbarton Sheriff Court on Saturday, John Leathem made no plea or declaration and was remanded in custody.
England should recall Kevin Pietersen for the forthcoming World Twenty20, according to Michael Vaughan.
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Hundreds of people gathered at a Clydebank park on Saturday night to remember teenager Paige Doherty.