Both RR Donnelley, which has a turnover of around $2.5bn, and $675m-turnover Bowne, have signed a definitive agreement for the all-cash deal, which is valued at around $481m, or $11.50 per share.
The agreement has been approved by both companies' boards of directors, but is still subject to regulatory approval and the consent of Bowne shareholders.
RR Donnelley chief executive Thomas Quinlan said: "Bowne is an exceptional fit with RR Donnelley. This combination satisfies all of the strategic imperatives that we evaluate as we consider acquisitions."
The US-based global print giant last year made an offer for the assets and properties of failed Canadian printer Quebecor World, which was later rescinded.
Both RR Donnelley and Bowne have UK-based divisions, with turnovers of £280m and £20m respectively.