Administrators Paul Stanley and Dean Watson, of Begbies Traynor, were appointed to the Aintree, Liverpool-based business earlier this month (8 January).
"Unfortunately, the business had been loss-making for a couple of years and had run out of funding," Watson said.
Callprint financial director Jonathan Drew said that the company would look to build on Redwood's environmental brand with its own digital print offering.
"This will enable the enlarged group to deliver an enhanced print and reprographics solution to all its clients," he added.
Callprint, which has expanded its litho capacity and presence in the North West with the buy, plans to retain Redwood's existing clients and grow its sales.
Managing director of Redwood, Jon Aindow, who has been in the print industry for more than 20 years, was unavailable for comment.
Redwood was established on 1 January 2008 as an environmentally friendly printer. Callprint has 220 staff, 21 branches and an annual turnover in excess of £16m.