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By Berkeley Lovelace, Jr., Special to

Walt Disney is said to be working with an adviser on a potential Twitter bid, according to a report on Monday, and the speculation sent the social network's shares on an intra-day run.

In recent months, speculation that Twitter will be sold has gathered steam. Last week, sources told CNBC that several technology and media companies had expressed interest in acquiring Twitter, including Google and Salesforce.

Earlier in the day, Twitter's stock was down 4 percent but turned positive after the news, with trading almost double its 30-day average volume. Meanwhile, Disney's shares were down more than 1 percent after the news.

"As a matter of policy, we don't comment on rumors or speculation," Twitter said in a statement.

Twitter CEO Jack Dorsey is a board member at Disney, according to the entertainment giant's website.