General manager Trent Baalke also might use one of his 13 picks in next week's draft to bolster the position for the NFC champions, who regularly use two tight end sets with Vernon Davis and another on the field.
Morrah, who played collegiately in the Bay Area at California, was a seventh-round draft pick by the division rival Seattle Seahawks in 2009. He has 16 career catches for 194 yards in 27 games, with seven starts.
San Francisco picks 31st overall, and Stanford tight end Zach Ertz could be in the mix if still available. He attended the team's pro day Wednesday.