“Everything is in order. Real Madrid have worked out his contract and there are no problems,” the Spanish football federation stated.
Solari has led Real to four wins out of four since being installed on a caretaker basis following Lopetegui's abrupt sacking last month.
And with 15 goals scored and only two conceded he has established the best ever start for a Real manager.
After thrashing Melilla in the Copa del Rey, they won 5-0 away to Viktoria Plzen in the Champions League, between victories over Real Valladolid and Celta Vigo in La Liga.
The upturn has left Madrid only four points adrift of league leaders Barcelona and on the brink of qualification for the last 16 in Europe.
Solari, who spent five years playing for Real, was promoted from his role in charge of Castilla, the club´s B team, after Lopetegui had overseen a miserable run that included just one victory in seven games.
The last of those, a 5-1 thumping at the hands of Barcelona, proved the final straw for club president Florentino Perez but the team have rebounded under their new coach.
Solari is Perez's 13th manager in 15 seasons across two seperate periods at the Santiago Bernabeu.
“He has done excellently,” Madrid club director Emilio Butragueno said on Sunday night. “We are very happy with how things have gone.”
Few could argue with the results, which have now earned Solari the chance to prove himself over a longer period.
He indicated in his opening press conference that he was open to keeping the job full-time.
“It's a great opportunity and it's a great club,” Solari said. “I don't just say that now as coach of the first team. I've been here with other teams, I've played and sweated in the shirt of this club.
“Madrid is bigger than all of us but it touches us all with its greatness. I want to be a part of it.”
— MARCA (@marca) November 12, 2018
Zinedine Zidane also stepped up mid-season from coaching Castilla in 2016 and led the team to Champions League glory five months later.
Solari has played down the comparisons with Zidane, who commanded instant respect owing to his glittering playing career and long-established status at the club, enhancing his status as a Real legend by winning three consecutive Champions League crowns.
“He's one of the greatest things we've ever had at this club,” Solari said last month. “He's very calm, a great coach and nobody can compare to him.”
The challenge for Solari will be to manage the dressing room when results take a turn for the worse.
After the international break, Madrid´s generous run of games continues with league matches against Eibar, Valencia, Huesca, Rayo Vallecano and Villarreal before Christmas, none of whom currently reside in La Liga´s top half.
They will also play Roma in Italy and CSKA Moscow at home in the Champions League, two games likely to settle who goes through top of Group G.
Solari was only allowed two weeks as an interim coach according to Spanish regulations, meaning Sunday´s trip to Celta Vigo was always likely to precede a decision on his future.
It proved an impressive final audition as Madrid struck a 4-2 victory, with Solari having to adapt to a handful of injury problems during the game.
“We´ll go to the death with him, he's a great coach and of course we want him to continue,” Madrid defender Alvaro Odriozola said afterwards.
This will be Solari´s maiden job in charge of a first team.
Antonio Conte, Mauricio Pochettino, Jose Mourinho and Joachim Low had all been linked to the position, with fresh approaches still possible next summer.