The draw paired Barca with United in a repeat of the 2009 and 2011 finals, both of which were won by the LaLiga giant under now Manchester City boss Pep Guardiola.
An injury-hit United overturned a 2-0 first-leg home loss to knock out Paris Saint-Germain as the Premier League side continues to thrive under Ole Gunnar Solskjaer.
But it lost a second straight match on Sunday (AEDT) when Wolverhampton Wanderers ended its FA Cup run at the quarter-finals stage.
Barca, meanwhile, thumped Lyon 5-1 to move into the last eight for a record 12th consecutive season, having been held 0-0 by the Ligue 1 team in the first leg.
Next month's double-header promises to be an enticing affair, with Valverde's side on track for a treble, sitting clear at the top of the LaLiga table and having already booked a Copa del Rey clash with Valencia.
But Valverde will not be taking anything for granted when Barca come up against United, which is fifth in the Premier League after a 2-0 loss at Arsenal in its previous league match.
"Difficult, but we knew any [draw] would be in the quarter-final of the Champions League. All teams have high hopes and fully deserve to be where they are," Valverde said of the draw, which will see the first leg played at Old Trafford.
"United are a historic European, world team, as we are, so it's an attractive tie and difficult. They're stringing together good results at the moment, so it's a very open tie that either side can win.
"They did well with a great result away to PSG, so we have to be careful. They do not give up and they were carrying a lot of injuries [against PSG], which they probably won't against us. It's one to look forward to."
Barca plays away to Real Betis on Monday (AEDT) in its final game before the March international break.