Citing "several people briefed on the matter", The New York Times has claimed that Jack Dorsey, Twitter's co-founder and an investor in Instagram, pushed a move to purchase the start-up in the months ahead of the Facebook deal.
Dorsey was one of the earliest backers of Instagram, the mobile app that allows iOS and Android users to apply filters to photos and then share them with other users.
He was previously a heavy user of Instagram, sending images to his Twitter feed every day, but he has not used the service since the deal with Facebook was announced.
In a lengthy report on the origins of the acquisition, the newspaper suggests that this is "perhaps a sign that he is not happy to see it in the hands of a competitor".
A Twitter spokeswoman declined to comment on the report.
Even if he is upset that Twitter missed out on Instagram, Dorsey is set to personally benefit from the deal, as he will get a healthy return on his investment in the start-up.
A funding round earlier in the month valued Instagram at around $500m, but it is thought that Twitter's rumoured interest in the firm may have pushed the price up to an eye-watering $1bn, raising fears of a new technology bubble.
However, Instagram is growing at an impressive rate, adding 10m new users following its launch on Android and acquisition by Facebook. It now has 40m members and is estimated to be adding around 1m more a day.