The former Tottenham Hotspur defender tweeted in response to Walker's post on Tuesday morning.
Spurs finished fifth last term and added Burnley's Kieran Trippier to their ranks, putting Walker's future at the club in doubt.
Walker, though, has brushed aside the 25-year-old's challenge to feature in all but five of the Londoners' Premier League fixtures so far.
The former Sheffield United defender's displays helped Mauricio Pochettino's side mount an unlikely title challenge, which was finally curtailed on Monday as Spurs were held to a 2-2 draw by Chelsea.
The following morning Walker tweeted his thanks to the Tottenham fans for their support throughout the campaign, before congratulating Leicester on their title success.
...our aim now is to secure 2nd place and the club's highest finish for 53 years. Lastly, many congratulations to @LCFC, amazing achievement— Kyle Walker (@kylewalker2) 3 May 2016
And Roberts, who retweeted both of Walker's posts, was quick to praise the England international.
@kylewalker2 fantastic words from the most improved player of the season well done kyle.— Graham Roberts (@GrahamRoberts4) 3 May 2016
"Fantastic words from the most improved player of the season", he posted in reply to Walker. "Well done Kyle."
Walker's teammates Eric Dier and Erik Lamela were far from guaranteed starters this time last year, and their impressive displays this season only add weight to Roberts' claim.
At the time of Trippier's arrival, the Daily Mail reported that Spurs would consider offers for their number 2.