Webber has been in the role for five years, having worked in the corporate sector for a quarter of a century. His previous roles included leading comms and public affairs at Merrill Lynch and GE.
Hill, currently head of corporate affairs and sustainability, will step up when Webber departs in December.
Before joining the company in 2015 as head of sustainability, she spent several years at PwC, rising to COO, sustainability and climate change.
In an internal memo, Webber said: "It's been an absolute privilege to have led such an outstanding team. From rebranding the business to establishing Landsec as Europe's leading sustainable real estate company, I'm proud of what we have achieved. I know, thanks to the talent across the business, that the future for Landsec is more exciting than ever, but after a quarter of a century in the corporate world, the time has come to take a more plural role and pass the baton on."
Landsec, which specialises in office, retail and mixed-use properties, has a portfolio that includes London sites New Street Square (EC4), Cardinal Place (SW1) and One New Change (EC4), alongside developments outside the capital such as Bluewater in Kent, Trinity Leeds and Gunwharf Quays in Portsmouth.
The company, which is listed on the London Stock Exchange, last week reported a pre-tax loss of £147m for the first half of the year. It cited the "challenging" retail market, "with retailers facing structural change, economic pressures and a rising cost base".
It also pointed to "further high-profile CVAs and administrations in the period", naming Debenhams and Arcadia.