According to reports in Germany, the Red Devils have pulled together a list of players whom they could target at left-back and centre-forward.
But talkSPORT understands that transfer funds are at a premium in January, meaning that new signings could be tricky to pull off.
With Luke Shaw and Tyrell Malacia struggling for fitness, a left-sided defender is seen as a top priority for new boss Amorim.
Meanwhile the addition of a proven, prolific centre-forward is also a focus with United desperate to add goals to their ranks.
Randal Kolo Muani is one player linked, with the striker looking for a move away from Paris Saint-Germain in January, just 18 months on from his £76million move to the Ligue 1 giants.
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It's reported that he could be available on loan or on a permanent basis as he looks to kickstart his career away from the Parc des Princes, where he has found opportunities limited despite the exit of Kylian Mbappe to Real Madrid.
Muani was replaced at his former club Eintracht Frankfurt by Omar Marmoush who has really shone in the Bundesliga - he is also reported to be on United's watch list
A tally of 11 goals and seven assists puts him second in the list of top goalscorers in the German top flight behind Harry Kane.
According to Sky Germany, a move for him would cost in the region of £50-£60m.
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Another Bundesliga striker on the list is Victor Boniface who has been integral to the success of Xabi Alonso's Bayer Leverkusen.
But a move for Boniface would also cost big money with the 23-year-old Nigerian attracting major interest from around Europe.
Victor Osimhen is the final striker on the list and could be available in January.
The Nigerian has scored seven goals in seven league games while on loan with Galatasaray, but holds a £75million release clause in his Napoli deal that comes into effect in January.
At left back, Alphonso Davies is reportedly the preferred candidate for Manchester United.
The Canadian's contract with Bayern Munich expires next summer and with no sign of a new deal on the horizon, it seems increasingly likely that he'll move on.
AC Milan star Theo Hernandez is also a target, having proven himself as one of the best attacking wing-backs in Europe during his time at the San Siro.
Milos Kerkez is the final name on the list after talkSPORT reported United's interest in the summer.
The Hungarian left-back has played in all 12 games for Bournemouth this season and could be seen as a long-term solution to the club's left back problem.
Whether Man United will be able to strengthen remains to be seen though, with transfer funds expected to be somewhat limited going into the January transfer window.