And the midfielder came closest to breaking the deadlock for us in the first half as we enjoyed the best of the action in our final reserve outing.
But Ipswich, who needed a win to secure the FA Premier Reserve League title, also had their chances and hit the woodwork twice in a minute at the start of the second half.
The first half had anything but an end-of-season feel but goalmouth action was at a premium.
Wayne Brown somehow headed over his own bar from Matt Etherington's third-minute cross and John Sutton's speculative volley from 30 yards forced a low save from the scrambling Mike Salmon.
The lively Etherington provided the threat again after 20 minutes when he skipped around William Snowden but his cross from the byline just skimmed Sutton's head.
Redknapp almost capped his competitive debut with 38 minutes on the clock with a volley worthy of the best move of the match. George Snee clipped in a clever pass to Sutton, he cushioned his header into the path of Redknapp whose first-time volley flew inches wide.
But Ipswich should have grabbed the lead just before the break. Darren Ambrose slid the ball into Ian Westlake, in space, 10 yards out, only for Lars Hirschfeld produced a fine one-handed save.
The visitors struck the woodwork twice in a minute at the start of the second half. With 52 minutes gone Richard Naylor struck the post with Hirschfeld stranded and seconds later Ambrose's looping header bounced back off the crossbar.
We responded on the hour when Gary Doherty's cross found Sutton whose second attempt was blocked bravely by Ashley Nicholls. Salmon then tipped aside John Piercy's low drive.
Redknapp tried his luck again after 70 minutes but curled his free-kick a yard wide, Leonhardsen blazed over from the corner of the area and Doherty was denied by a fine Salmon save after another Redknapp pass.
Redknapp left procedings after 80 minutes, Marney, Jackson and Piercy all shot wide as we continued to push on but the goal wasn't forthcoming.
Spurs Reserves: Hirschfeld (Burch, 74), Kelly, Jackson, Piercy, O'Donoghue, Doherty, Leonhardsen, Snee, Sutton (Henry, 62), Etherington, Redknapp (Marney, 80). Unused subs: Barnard, Hughes.
Ipswich Town Reserves: Salmon, Artun, Snowdon, Nicholls, Miller, Brown (Beevors, 64), Ambrose, Bloomfield, Naylor, Counago (Richards, 72), Westlake. Unused subs: Logan, Price, Robinson.
Jamie Redknapp completed 80 minutes as our reserves had the better of a goalless draw against title-chasing Ipswich Town on Monday night.