The 27-year-old has been a rumoured target for the Red Devils for some time after impressing in Serie A with the Rome side.
United are believed to be attracted to his versatility, with the Romanian able to fill in at centre-back as well as his preferred role on the left side of the backline.
But despite their manager-less status, Spurs are also now in the hunt.
Chairman Daniel Levy and sporting director Franco Baldini are determined not to let the search for a new head coach hold up transfer plans, and are keen to lure Radu to London in the coming months.
The 14-times capped enforcer may well be tempted by Premier League life, with Lazio having only managed a ninth-placed finish in the Italian top tier last term.
United are likely to move if Tottenham attempt to beat them to Radu, with van Gaal looking to bring in a replacement for veteran left-back Patrice Evra as a part of his predicted mega spend.