According to media reports, the microblogging website has finalised a cash and shares deal with Dodsworth, which is expected to be officially confirmed today.
Twitter has remained quiet on the speculation, which first emerged last month, and issued a statement on Monday saying that it will "continue to not comment on rumours".
TweetDeck's software enables users to organise their various Twitter accounts from a single launchpad, while also integrating data from other social networks such as Facebook.
Dodsworth, a programmer who has worked at the Prudential and PricewaterhouseCoopers, founded TweetDeck in 2008 after becoming frustrated at missing messages from friends.
The company employs 15 staff and is based in the Old Street area of London, dubbed the 'Silicon Roundabout' due to the large amount of technology startups.
Despite TweetDeck failing to ever turn a profit, the company has received the support of many high profile Twitter users, such as Stephen Fry and Ashton Kutcher.
Twitter's interest in TweetDeck accelerated after reports emerged that internet entrepreneur Bill Gross's UberMedia company had submitted a $30m offer.
Twitter is currently searching for ways to unlock its massive user base to generate revenue streams, including the launch of "promoted tweets" that appear when users submit searches. The company has also allowed ads to appear on TweetDeck under a revenue-share agreement.
On Saturday, Twitter registered its highest ever volume of UK traffic as users rushed to the microblogging site to discover the identity of the footballer who had taken out an injunction.