A feasible strategy would be to invade the Middle East. Before you rip this plan apart consider the following. Now to be honest, I can’t assure anyone that a Med/M. East strategy would be sufficient for victory, but it would have certainly been less costly than a 41’ Russian invasion-- any analysis shows that. Compared to the 153 divisions needed to invade Russia and the differential in potential casualties it was a more cost effective strategy. With Germans to the West and South the political balance of power in Turkey could have been turned in Germany's favor (particulary cut-off from Allied-non Russian- support). This would put German in a much better position vis-a-vis a potential invasion of Russia. With a German-Turkish Army group poised to invade the southern Causasus and three more facing Russia's western borders success was more likely and a political solution of co-operation with Hitler by Stalin also possible (at least the threat of attack by Stalin against German held territories less likely). With an intact German army & Luftwaffe (that is neither decimated in a Russian invasion) it is not likely the Western allies could achieve battlefield success in north Africa nor in any 'return to the continent' (208 + German divisions would have been quite a deterrent even if half had to face the Russians).
All that being said Germany would probably have been better off pursuing such a strategy. It could have severely weakened Britain and been in a position to attack Russia from the south as well as the west (though the Turkish rails could not have supported a large force it could have supported a force adequate to take the Caucasian oil fields). It certainly would have been a great defeat for the British and would have enhanced Germany's strategic position. Iraq, Persia and Vichy run Syria would have been pro-Axis and would have been strategically useful. For example, the significant Lend Lease aid to Russia which went via Persia wouldn't have occurred. Probably the main benefit would have been the capture of the Suez Canal. This would have had a great effect on British morale. The capture of Egypt & Suez might have influenced Franco to allow Germany to capture Gibraltar & then expel the British from the Mediterranean altogether, thus removing the later threat of Italy's invasion.
The forces available are: 15-20 German Divisions inc. 3 armoured (For example, at the time of rea? Barbarossa 38 German divisions were in the "West", 12 in Norway and 7 in the Balkans uncommitted to the attack on Russia ? with no need to maintain this level of forces in situ - no British threat, no invasion in France. Also no invasion and so no need to garrison Yugoslavia or Greece at least 20 divisions could have been made available for an offensive on the USSR via Iran and Turkey); 25 Italian Divisions inc. 3 armoured (partially re-equipped with captured British tanks), would also inc. all Alpine troops; Strategically it would not have required a large force. Rommel's German forces at full strength with four mobile divisions, three infantry/parachute brigade equivalents, three artillery brigade/group equivalents not to mention the Italian forces would have been more than adequate.
This is especially true if they had been able to galvanize the Italians to make good use of their navy. While it would require the 'grounding' of virtually ALL Italian vehicles the subsequent capture of transport and supplies would be adequate for short term sustenance of a drive on the Suez. If such a force were supported by the airborne forces available to Germany chances for success increase. Historically the RN made plans to evacuate the East Med once Rommel entered Egypt, no doubt they would have done so in '41 also. With the capture of supplies and transport, the excellent port/capacity of Alexandria drive into the Levant was possible. It is not likely Mussolini would have refused Hilter had the latter insisted on a Med strategy, particularly with the fiascos of Greece, Sidi Barrani and the ongoing debacle in E. Africa.
In his writings, Rommel certainly mused on the possibility of siezing the Middle east oil fields and stated that this would solve all of Germany's POL needs (see The Rommel Papers). He also developed an outline plan for taking the Nile if he had not been stopped at the 1st battle of El Alamein. He was known for his selective disobedience of his Italian superiors. who would have argued with success? '41 was the prime moment for pursuit of a Med strategy with the objective of a superior position for invasion of Russia. Commonwealth forces were at their weakest in N. Afr/M.East in relation to Axis strength. German airpower was at its greatest in relation to allied airpower. A Med. strategy with all its difficulties would be easier and less costly in men and equipment than a '41 Russian invasion. Loss of after loss would have a cumulative effect on the politics of Great Britain & Germany's chances for success much better that her other options.
Had Hitler pushed for it with all the diplomatic and military pressure and co-ercive options available it could have resulted in cooperation. Capturing Alexandria would have forced the RN to abandon E. Med clearing the way for Italian and occupied country shipping. As it was Rommel’s historical force was three armored divisions (DAK & Ariete) and one motorized division (Trento). Substituting a German panzer and motorized unit for the two historical Italian units is not just dropping four motorized divisions into the theater. With an entire Luftflotte in support, the airborne forces and if Turkey is successfully co-erced into cooperation it all becomes possible.